The polynomiality conjecture for $S_{k,j}(n)$

Let $t:=n(n+1)$ and define

$$Q_j(n):=\prod_{q=1}^{j-1}(jn+q).$$

Polynomiality conjecture. For the indicated integers $k$ and $j$, $S_{k,j}(n)$ is expressed as a polynomial in

$$Q_j(n)\mathbb{Q}[t].$$

This claim is suggested by the explicit formulas computed in the paper, but the source gives no proof or resolution.