Minimum number of integer-sided squares needed to tile an $m$ by $n$ rectangle.

Let $T(m,n)$ for integers $m,n$ be the least number of integer-sided squares needed to tile an $m\times n$ rectangle. Clearly $T(kx,ky)\leq T(x,y)$. Are there integers $x,y,k\gt 1$, such that $T(kx,ky)<T(x,y)$?

$T(m,n)$ is tabulated at the OEIS. Also, several references are given:

Richard J. Kenyon, Tiling a rectangle with the fewest squares, Combin. Theory Ser. A 76 (1996), no. 2, 272-291.

Mark Walters, Rectangles as sums of squares, Discrete Math. 309 (2009), no. 9, 2913-2921.

Bertram Felgenhauer, Tiling rectangles by minimal number of squares.

Maybe you could have a look at those, to see whether your question is considered (and, if it is, you could then report back).

I've posted a 4944 possible counterexamples at Possible Counterexamples to the Minimal Squaring Conjecture. Odds are at least one of these is a valid counterexample.

Data and verified pictures up to 388 at Minimally Squared Rectangles

Other information at tiling a rectangle with the smallest number of squares

According to my notes there, my minimal possible counterexample is currently

Here's the Bouwkamp code of two simple perfect squared rectangles:

• 13: 304x274 (141,78,85)(71,7)(92)(133,8)(51,28)(23,97)(74);
• 15: 152x137 (83,69)(31,38)(54,29)(16,15)(8,30)(25,4)(1,22)(21).

There is no lower-order simple squared rectangle of either size, but there might be compound ones. If not then T(304,274) = 13, T(152,137) = 15, and T(304,274) < T(152,137).