Conjecture on integer ratios of consecutive odd power sums

For positive integers $k$ and $m$, let

$$T_k(m):=\sum_{j=1}^{m}(2j-1)^k.$$

Conjecture on odd power sums.

$$\left\{\left.\frac{T_k(m+1)}{T_k(m)}\,\right|\, k,m\in\mathbb Z^+,\ m>1\right\}\cap\mathbb Z=\varnothing.$$

Equivalently, there are no positive integers $a$, $k$, and $m>1$ satisfying $aT_k(m)=(2m+1)^k$. The conjecture is motivated by the paper's unsolvability results for several classes of $a$ and by its stated theorem, but no resolution is supplied in the text.