The nonintegrality 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

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

Nonintegrality conjecture. For any $r>s>1$, there is a positive integer $n$ such that $c_n^{(r,s)}$ is not an integer. Together with the finite integrality conjecture, this proposes a contrast between finite initial integrality for suitable $r$ and eventual failure of integrality for every fixed pair $r>s>1$. The source presents both assertions as open arithmetic problems.