Miana–Ohtsuka–Romero's Catalan triangle cubic-sum identity

Let $B_{n,k}:=\frac{k}{n}{2n\choose n-k}$ for $n\geqslant k\geqslant 0$, and let $r=\min\{n,m\}$ and $s=\max\{n,m\}$. Miana–Ohtsuka–Romero's conjecture. For all positive integers $m$ and $n$, the identity

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

holds. In the particular case $m=n$, this recovers the previously established identity for $\sum_{k=0}^n B_{n,k}^3$. The paper proves a stronger version, since the identity holds separately for $r=m,s=n$ and for $r=n,s=m$.