What numerical lower bound on the index of an odd perfect number is implied by the results in F.-J. Chen and Y.-G. Chen's 2014 paper?

The smallest possible value of the index $m=\sigma(N/q^\alpha)/q^\alpha$ when $q=p$ is $$3^3\times 5^3=3375$$

Proof :

Let $M$ be the number of prime divisors counted with multiplicity.

Using $M$, we can classify the $30$ forms as follows :

$$\small\begin{align}M=1&:q_1 \\\\M=2&:{q_1}^2, q_1 q_2 \\\\M=3&:{q_1}^3, {q_1}^2 q_2,q_1 q_2 q_3 \\\\M=4&:{q_1}^4, {q_1}^3 q_2,{q_1}^2 {q_2}^2,{q_1}^2 q_2 q_3,q_1 q_2 q_3 q_4 \\\\M=5&:{q_1}^5, {q_1}^4 q_2, {q_1}^3 {q_2}^2,{q_1}^3 q_2 q_3,{q_1}^2 {q_2}^2 q_3,{q_1}^2 q_2 q_3 q_4,q_1 q_2 q_3 q_4 q_5 \\\\M=6&:{q_1}^6,{q_1}^5 q_2, {q_1}^4 {q_2}^2,{q_1}^4 q_2 q_3,{q_1}^2 {q_2}^2 {q_3}^2,{q_1}^3 q_2 q_3 q_4,{q_1}^2 {q_2}^2 q_3 q_4,{q_1}^2 q_2 q_3 q_4 q_5,q_1 q_2 q_3 q_4 q_5 q_6 \\\\M=7&:{q_1}^7, q_1 q_2 q_3 q_4 q_5 q_6 q_7 \\\\M=8&:{q_1}^8\end{align}$$

For $M\le 5$, the $18$ forms are all the possible forms, so there is no form that $m$ can take.

For $M=6$, we see that ${q_1}^3{q_2}^3,{q_1}^3{q_2}^2 q_3$ are not included, so the smallest possible value of $m$ is $3^3\times 5^3=3375$.

For $M=7$, the smallest possible value of $m$ is $3^6\times 5=3645$.

For $M=8$, the smallest possible value of $m$ is $3^7\times 5=10935$.

For $M\ge 9$, the smallest possible value of $m$ is $3^9=19683$.

Therefore, it follows that the smallest possible value of the index $m=\sigma(N/q^\alpha)/q^\alpha$ when $q=p$ is $$3^3\times 5^3=3375$$