Odd Catalan-triangle power congruence conjecture

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

$$\sum_{k=1}^n B_{n,k}^{2r+1}\equiv {2n-1\choose n}\pmod{{2n\choose n}}$$

if and only if $n=2^a-2^b$ for some $0\leqslant b<a$. This gives a binary characterization of the indices for which the stated congruence holds; the paper presents computational examples but no proof of the general assertion.