Mixed Catalan-triangle product divisibility conjecture

Let $B_{m,k}$ and $B_{n,k}$ denote the Catalan-triangle quantities used in the paper. Let $m,n,s,t\geqslant 1$ and $r\geqslant 0$, with $r+s+t\equiv 1\pmod 2$. Mixed-product divisibility conjecture. The sum

$$\sum_{k=1}^n k^r B_{m,k}^{s}B_{n,k}^{t}$$

is divisible by

$$\frac{1}{2}\frac{(2m)!(2n)!}{m!n!(m+n)!}.$$

The statement is presented as a generalization of an earlier corollary for products of Catalan-triangle terms. Its general validity remains open in the source.