The equality of the generalized sums $\tilde{S}_{k,j}(n)$ and $S_{k,j}(n)$

Let $x_r:=r^{2k+1}$, let $C_{j,q}(n)$ be the coefficients defined by the paper, and define

$$s_{k,j}(n):=\sum_{q=0}^{j(n-1)}C_{j,q}(n)x_{j+q}.$$

For integers $j\geq 1$, let

$$\tilde{S}_{k,j}(n):=\sum_{\{\lambda\}\in B_{j,jn}}\left\{\lambda_1^{2k+1}+(\lambda_2-n)^{2k+1}+\cdots+(\lambda_j-jn+n)^{2k+1}\right\},$$

where $B_{j,jn}:=\{\lambda_k:1\leq\lambda_1\leq\cdots\leq\lambda_j\leq jn\}$, and set

$$S_{k,j}(n):=\sum_{q=0}^{j-1}{j(n+1)\choose q}s_{k,j-q}(n).$$

The generalized-sums equality. For any $k\geq 0$ and $j\geq 1$, one has

$$\tilde{S}_{k,j}(n)=S_{k,j}(n).$$

The equality is proved in the paper for the particular case $j=2$, while the general case is presented as a conjecture.