Even-exponent Catalan-triangle congruence conjecture

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, and let $n,r,s\geqslant 1$. Even-exponent Catalan-triangle congruence conjecture.

$$\sum_{k=1}^{n}k^{2r}B_{n,k}^{2s+1} \equiv \begin{cases} \displaystyle {2n-1\choose n},&\text{if $n=2^a-1$,}\\[5pt] 0,&\text{otherwise} \end{cases} \pmod{{2n\choose n}}.$$

This is a complementary parity case to the preceding Catalan-triangle congruences, asserting that only indices one less than a power of two give the nonzero residue. The source supplies no resolution.