Conjecture equating integer ratios for ordinary and odd power sums

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

$$S_k(m):=\sum_{j=1}^{m-1}j^k,$$

and

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

Equality conjecture for power-sum ratios.

$$\left\{\left.\frac{S_k(m+1)}{S_k(m)}\,\right | k,m\in\mathbb Z^+, m>3\right\}\cap\mathbb Z=\left\{\left.\frac{T_k(m+1)}{T_k(m)}\,\right | k,m\in\mathbb Z^+, m>1\right\}\cap\mathbb Z.$$

The assertion proposes that the sets of integer ratios in the ordinary and odd power-sum problems coincide. The supplied text gives no resolution status.