Integer-valued Delannoy-polynomial power conjecture

Let $d_k(x)$ and $s_k(x)$ be the two Delannoy-related polynomials considered in the paper. Integer-valued Delannoy-polynomial power conjecture. For all positive integers $m$ and $n$, both polynomials

$$\frac{1}{n}\sum_{k=0}^{n-1}(2k+1)d_k(x)^m s_k(x)^m$$

and

$$\frac{1}{n}\sum_{k=0}^{n-1}(-1)^k(2k+1)d_k(x)^m s_k(x)^m$$

are integer-valued. This conjecture is based on numerical calculations and extends the paper's proved integer-valued results for particular even powers. The source gives no proof or resolution.