The finite integrality conjecture for Schmidt-type coefficients

For positive integers $r$ and $s$, let $c_k^{(r,s)}$ be the uniquely determined rational numbers, independent of $n$, satisfying

$$\sum_{k=0}^n{n\choose k}^r{n+k\choose k}^r=\sum_{k=0}^n{n\choose k}^s{n+k\choose k}^s c_k^{(r,s)}.$$

Finite integrality conjecture. For any $s>1$ and $n\geqslant 0$, there is an integer $r>s$ such that all the numbers $c_k^{(r,s)}$ for $0\leqslant k\leqslant n$ are integers. This asks for arbitrarily long initial segments of the coefficient sequence to become integral after choosing a suitable exponent $r>s$; the paper gives computational values for the least such $r$ when $s=2$, but does not resolve the conjecture in general.