Mixed cubic identity for Catalan triangle numbers

Let $B_{n,k}$ be the Catalan triangle numbers, let $n,m\in\mathbb{N}$, and set

$$r=\min(n,m),\qquad s=\max(n,m).$$

Mixed cubic identity. The identity

$$\sum_{k=1}^{r}B_{n,k}^{2}B_{m,k}=\frac{1}{2}\binom{2n}{n}^{2}\binom{2m}{m}\left[1-\frac{n+2m}{r}\binom{n+m}{n}^{-1}\binom{n+r}{n}^{-1}\sum_{j=0}^{r-1}\binom{s+j}{s}\binom{n+j}{n-1}\right]$$

holds. When $m=n$, it specializes to the identity stated in the cited corollary. The formula is presented as a conjectural identity, and the supplied context does not provide an analytic proof or a resolution.