Can product of distinct $\frac{3m+2}{2m+1}$ be an integer?

Friend gave me this problem (although I don't think he knows the answer). The question is whether the product $$ \frac{3m_1+2}{2m_1+1}\cdot\frac{3m_2+2}{2m_2+1}\cdots \frac{3m_n+2}{2m_n+1} $$ can be an integer for some distinct $m_i \in \mathbb{N}$ (here $0 \not\in \mathbb{N}$, otherwise we would have simple solution $n=1, m_1=0$, giving product $2$). So basically this is about products of combinations of numbers $$ \frac{5}{3},\frac{8}{5},\frac{11}{7},\frac{14}{9}, \frac{17}{11}, \frac{20}{13}, \frac{23}{15}, \frac{26}{17}, \frac{29}{19}, \frac{32}{21}, \dots $$

My try so far:

We can rule out fractions which have denominator divisible by $3$, since no numerator can be divisible by $3$, so basically $m \not\equiv 1 \pmod 3$. I have then tried computer checking all products with $m_i \in \{2,3,5,6,8,9,\dots,39\}$, none of them yield an integer.

There are solutions for many values of $n$. You can find here solutions for $n=5,\;\;7,8,9,10,\ldots, 21$.

One of solutions for $n=5$:

$$ (m_1,m_2,m_3,m_4,m_5)=(9,12,14,27,41): $$

$$ \dfrac{3\cdot 9+2}{2\cdot 9 + 1} \times \dfrac{3\cdot 12+2}{2\cdot 12 + 1} \times \dfrac{3\cdot 14+2}{2\cdot 14 + 1} \times \dfrac{3\cdot 27+2}{2\cdot 27 + 1} \times \dfrac{3\cdot 41+2}{2\cdot 41 + 1} = \\ \dfrac{\color{red}{29}}{\color{violet}{19}} \times \dfrac{\color{violet}{38}}{\color{blue}{25}} \times \dfrac{44}{\color{red}{29}} \times \dfrac{\color{green}{83}}{55} \times \dfrac{\color{blue}{125}}{\color{green}{83}} = \\ \dfrac{503063000}{62882875} = 8.$$

Another examples of solutions for $n=5$:
$(9,12,14,21,71) \rightarrow 8$;
$(6,12,27,41,47) \rightarrow 8$;
$(6,12,21,47,71) \rightarrow 8$;
$(9, 11, 12, 30, 101) \rightarrow 8$;
$(8, 9, 14, 71, 107) \rightarrow 8$;
$(6, 8, 47, 71, 107) \rightarrow 8$;
$(5, 12, 41, 47, 110) \rightarrow 8$;
$(6, 8, 39, 87, 131) \rightarrow 8$;
$(6, 11, 17, 99, 132) \rightarrow 8$;
$(3, 99, 123, 126, 132) \rightarrow 8$;
$\ldots$
$(3,30,1943,5351,5652) \rightarrow 8$;
$(3,30,2081,4371,5912) \rightarrow 8$;
$(3,30,2133,4680,5103) \rightarrow 8$;
$\ldots$

Now focus on $n\ge 7$. For each $n\ge 7$ we search integer solution $q, (m_1,m_2,\ldots,m_n)$ for equation $$ \prod_{j=1}^n \dfrac{3m_j+2}{2m_j+1} = q \tag{*}. $$

For each $n\ge 7$ there are many solutions of eq. $(*)$. Few examples with rather small values $m_j$:

  n     q       (m_1, m_2, ..., m_n)

  7     19      (3,5,11,23,35,53,297)
  7     19      (3,5,9,23,35,297,1551)
  7     19      (3,5,11,17,53,59,4743)
 ...    ...     ...

  8     26      (21,71,147,221,237,255,267,356)
  8     28      (3,8,27,32,41,47,71,107)
 ...    ...     ...

  9     40      (12,27,41,45,47,51,68,81,108)
  9     41      (3,21,47,71,137,147,221,227,341)
  9     43      (3,5,15,17,23,35,53,501,609)
  9     44      (3,5,8,11,12,27,41,201,237)
 ...    ...     ...

  10    58      (57,123,605,671,753,821,1004,1079,1145,1227)
  10    59      (21,27,47,71,207,297,305,311,327,491)
  10    61      (5,11,47,69,71,107,161,339,467,509)
  10    62      (3,27,29,41,47,71,107,137,144,311)
  10    64      (6,8,9,12,14,27,41,47,71,107)
  10    65      (3,5,11,15,23,35,53,57,605,671)
 ...    ...     ...

  11    88      (47,71,81,87,107,108,131,144,197,296,311)
  11    89      (21,27,41,47,71,137,147,201,221,237,563) 
  11    91      (5,21,23,71,177,201,297,389,417,447,671) 
  11    92      (8,15,23,35,39,45,51,53,68,87,131)
  11    94      (5,8,11,17,47,71,107,201,237,291,297)
  11    95      (3,5,21,47,71,147,221,297,305,327,491)
  11    98      (3,5,8,11,21,71,137,147,201,221,237)
  11    100     (3,5,8,11,15,20,23,35,53,227,341)
  11    104     (2,3,5,11,15,23,35,53,59,447,671)
 ...    ...     ...

  12    131     (47,107,123,291,297,311,467,563,579,587,963,1091)
  12    134     (9,21,71,164,227,297,333,341,389,417,447,513)
  12    136     (11,15,23,35,53,87,131,144,192,197,296,311)
  12    139     (3,21,39,71,87,131,137,147,161,221,339,509)
  12    140     (9,11,12,15,20,23,101,177,201,237,291,335)
  12    142     (3,11,15,23,35,53,72,357,381,795,801,1017)
  12    143     (3,11,15,23,27,35,53,87,131,207,311,671)
  12    145     (3,5,9,23,35,53,297,333,351,357,527,791)
  12    146     (3,5,11,15,23,35,53,237,501,563,845,924)
  12    148     (3,5,8,15,23,32,35,51,53,171,333,357)
  12    160     (2,3,5,8,11,15,20,23,35,53,227,341)
 ...    ...     ...

  13    199     (21,39,71,87,131,137,147,201,221,465,671,1079,1127)
  13    200     (21,27,41,71,87,131,197,212,296,311,333,357,536)
  13    202     (8,27,41,71,107,201,237,287,351,431,437,527,791)
  13    203     (9,23,35,47,53,71,107,297,333,351,357,527,791)
  13    205     (9,11,23,35,53,101,107,297,351,527,579,587,791)
  13    206     (5,21,27,41,47,71,201,237,287,297,431,869,1304)
  13    208     (6,8,21,39,71,87,131,147,221,237,255,267,356)
  13    211     (3,11,27,39,41,87,131,237,263,395,593,753,773)
  13    212     (9,11,12,15,20,21,23,71,101,177,201,237,335)
  13    214     (3,5,21,47,71,87,131,297,305,671,851,1131,1212)
  13    215     (3,5,21,27,41,137,147,201,221,237,501,609,759)
  13    217     (3,5,11,23,35,53,227,297,341,389,417,447,671)
  13    220     (3,5,11,15,23,35,53,87,131,144,197,296,311)
  13    224     (3,5,8,11,12,27,41,99,123,132,149,168,224)
 ...    ...     ...

  14    295     (51,137,147,171,201,221,389,417,423,447,671,821,971,1317)
  14    296     (51,77,116,137,147,171,212,221,275,311,413,620,731,1097)
  14    302     (9,21,71,107,159,201,237,239,359,563,620,705,845,855)
  14    304     (8,27,41,71,107,137,147,201,206,221,237,255,267,356)
  14    305     (9,17,21,71,137,147,201,221,237,291,297,333,345,1151)
  14    308     (8,21,27,39,41,71,80,87,131,137,147,201,221,237)
  14    310     (9,12,21,27,41,47,71,101,107,201,237,335,501,609)
  14    311     (3,21,71,137,147,201,221,237,287,291,431,647,971,1347)
  14    316     (3,8,39,87,131,137,147,221,227,341,389,416,417,447)
  14    320     (6,11,12,15,23,35,47,51,53,71,107,161,171,242) 
  14    322     (3,11,15,23,35,51,53,60,87,131,171,179,513,837)
  14    323     (3,11,15,23,27,35,41,53,137,147,221,237,311,563)
  14    328     (3,9,11,12,27,30,41,47,101,137,147,221,227,341)
  14    332     (3,5,9,12,27,41,47,71,107,144,159,201,239,359)
  14    344     (2,3,11,15,23,27,35,41,53,107,237,501,609,759)
  14    352     (2,3,5,11,15,23,35,53,87,131,144,197,296,311)
 ...    ...     ...

  15    448     (21,71,77,80,116,137,147,206,221,237,255,267,356,357,536)
  15    452     (8,71,87,107,131,197,201,237,291,296,333,357,489,759,795)
  15    455     (9,21,47,71,101,137,147,201,221,335,423,471,707,927,1061)
  15    458     (8,27,39,41,71,87,107,131,201,203,237,333,357,381,399)
  15    460     (9,12,21,71,101,137,147,201,221,237,291,333,335,444,465)
  15    464     (8,9,27,41,71,80,107,147,221,237,255,267,356,396,467)
  15    470     (9,11,15,17,23,35,53,237,291,297,311,333,357,536,563)
  15    472     (8,11,15,23,35,45,51,53,68,71,107,212,255,275,413)
  15    476     (3,11,27,41,47,71,107,147,221,237,255,267,356,357,536)
  15    481     (3,11,15,23,35,51,53,137,171,333,357,383,575,863,1295)
  15    484     (3,11,15,23,27,35,41,53,164,237,287,344,431,759,795)
  15    485     (3,11,15,17,23,35,53,71,107,237,291,333,705,857,1131)
  15    488     (3,9,12,21,27,32,41,71,87,131,197,296,311,333,467)
  15    490     (3,5,8,21,71,137,147,201,221,227,237,341,471,707,1061)
  15    496     (3,5,8,27,32,41,47,63,84,95,143,164,215,323,485)
  15    508     (3,5,8,11,15,20,23,35,53,71,107,465,1079,1145,1227)
  15    512     (3,6,9,11,12,14,21,27,41,62,71,99,117,132,176)
 ...    ...     ...

  16    676     (21,27,41,71,80,137,147,221,237,255,267,356,383,575,863,1295)
  16    680     (9,21,71,137,147,201,221,237,333,395,489,563,591,593,674,753)
  16    682     (9,15,51,107,171,305,327,333,467,491,501,579,587,753,795,1004)
  16    686     (8,21,39,71,87,131,137,147,201,221,237,287,291,431,759,795)
  16    688     (8,27,39,41,71,80,87,131,147,221,237,255,267,356,396,467)
  16    692     (8,9,39,71,87,107,131,201,237,291,333,335,467,521,1092,1095)
  16    700     (9,15,23,27,35,41,51,53,80,137,147,171,221,333,396,467)
  16    704     (9,12,23,27,35,41,47,53,71,80,107,110,203,207,246,311)
  16    710     (9,11,12,15,23,35,51,53,171,227,341,389,417,447,449,671)
  16    712     (9,11,12,15,23,35,51,53,87,131,171,207,276,333,375,500)
  16    716     (3,23,27,35,41,47,53,57,71,86,107,227,297,341,389,417)
  16    728     (3,11,15,17,23,35,53,71,107,237,291,333,444,465,620,632)
  16    736     (3,5,8,45,51,68,77,116,135,137,180,240,383,575,863,1295)
  16    742     (3,5,8,21,27,41,71,137,147,201,221,237,287,431,647,971)
  16    752     (3,5,8,15,23,32,35,51,53,57,171,333,357,536,605,908)
  16    760     (3,5,8,11,15,23,32,35,53,87,131,171,669,671,717,1076)
 ...    ...     ...

  17    1024    (8,47,71,77,107,116,141,144,188,192,212,216,227,288,341,384,512)
  17    1028    (9,21,35,53,71,87,131,197,296,311,333,357,449,609,812,1091,1113)
  17    1034    (8,27,39,41,47,71,87,107,131,147,201,221,467,579,609,759,795)
  17    1036    (9,21,23,35,53,71,80,137,147,221,275,297,381,413,617,620,801)
  17    1040    (8,15,39,45,51,68,81,87,108,131,164,227,341,389,417,447,671)
  17    1048    (9,15,23,27,35,41,53,80,101,137,147,221,237,396,563,567,756)
  17    1052    (6,11,23,35,47,53,71,107,171,297,389,417,447,669,671,759,1139)
  17    1054    (9,11,12,21,27,71,101,201,207,237,311,335,437,501,609,881,1161)
  17    1060    (9,12,15,21,23,35,51,53,71,171,287,333,431,435,513,653,684)
  17    1064    (9,11,12,21,23,35,53,71,87,131,197,296,297,311,333,375,500)
  17    1072    (3,23,27,35,41,47,53,71,107,144,192,201,203,246,287,431,647)
  17    1088    (3,15,23,27,35,41,47,53,57,71,86,107,110,144,192,207,311)
  17    1096    (3,9,12,21,27,41,53,71,80,159,239,249,332,359,539,809,867)
  17    1100    (3,11,12,15,23,35,51,53,60,171,201,237,287,291,431,647,971)
  17    1136    (3,5,8,11,15,23,35,53,71,107,237,333,357,416,462,867,1301)
 ...    ...     ...

  18    1544    (8,21,39,71,87,131,147,201,221,237,255,267,356,396,423,467,471,707)
  18    1552    (8,11,71,99,107,123,132,137,147,171,201,221,237,333,566,705,857,1131)
  18    1568    (8,11,21,39,71,87,131,137,147,201,206,221,237,255,267,356,375,500)
  18    1580    (8,15,23,27,35,41,51,53,80,137,147,171,221,227,341,389,417,447)
  18    1592    (6,8,27,39,41,47,71,87,107,131,137,147,201,221,465,671,1079,1127)
  18    1600    (9,12,15,23,27,35,41,47,53,71,101,107,126,212,237,333,396,416)
  18    1616    (9,11,12,15,21,23,35,53,71,80,101,137,147,221,237,396,416,437)
  18    1664    (6,8,9,12,14,21,27,39,41,71,87,131,147,221,237,255,267,356)
  18    1696    (3,5,11,12,15,23,35,51,53,171,237,263,282,291,333,357,647,971)
 ...    ...     ...

  19    2336    (6,15,41,47,137,147,164,221,227,237,276,341,389,417,447,501,563,845,924)
  19    2366    (9,15,23,27,35,41,51,53,80,137,147,171,221,237,263,333,395,593,753)
  19    2368    (6,8,39,51,77,87,116,131,137,147,171,212,221,275,311,413,620,731,1097)
  19    2380    (8,11,15,23,35,51,53,71,107,171,179,237,287,431,759,795,821,837,1116)
  19    2384    (8,11,12,27,41,47,71,87,107,131,197,201,237,291,296,311,335,467,1092)
  19    2392    (9,12,15,21,23,35,53,71,80,81,101,108,237,335,423,471,707,927,1061)
  19    2432    (9,12,14,15,23,27,35,41,45,51,53,68,83,125,222,333,357,536,563)
  19    2560    (3,5,8,11,15,23,35,53,87,131,144,192,197,296,311,396,467,977,1466)
 ...    ...     ...

  20    3500    (9,21,23,35,53,71,80,137,147,221,227,297,341,389,417,447,671,821,971)
  20    3520    (8,15,39,45,51,68,81,87,108,117,131,176,185,237,278,287,291,431,759,795)
  20    3584    (6,11,24,27,41,47,69,92,99,123,126,132,147,221,237,255,267,356,357,536)
 ...    ...     ...

  21    5488    (6,11,15,17,21,23,35,53,71,80,99,132,137,147,171,221,237,471,669,707,1061)
  21    5504    (9,12,14,15,23,27,35,41,45,51,53,57,68,80,86,137,147,221,237,291,759)

Note 1: if $n\ge 8$, then eq. $(*)$ has solutions for different integer $q$.

Note 2: this integer product is divisible by $2$, $4$, $8$ in many cases (depends on parity of numbers $m_j$); solutions with odd $q$ exist too, but are much more rare.