Odd-exponent Catalan-triangle congruence conjecture with a power-of-four threshold

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, let $r\geqslant 0$ and $s\geqslant 1$, and assume $n\geqslant 4^s-1$. Power-of-four threshold conjecture.

$$\sum_{k=1}^{n}k^{2r+1}B_{n,k}^{2s} \equiv \begin{cases} \displaystyle {2n-1\choose n}4^{s-1},&\text{if $n=4^s-1$ or $n=2^a+1$,}\\[5pt] 0,&\text{otherwise} \end{cases} \pmod{{2n\choose n}4^{s-1}}.$$

The claim refines the preceding congruence patterns for powers of the Catalan triangle, with exceptional indices described by powers of two. Its general status is left open in the source.