Triple Catalan-triangle product divisibility conjecture for rows $n$, $2n$, and $3n$

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper. For positive integers $r,s,t$ satisfying $r+s+t\equiv 1\pmod 2$, triple-product divisibility conjecture. The sum

$$\sum_{k=1}^n B_{n,k}^{r}B_{2n,k}^{s}B_{3n,k}^{t}$$

is divisible by both

$$\frac{1}{3}{6n\choose n}$$

and

$${6n\choose 3n}.$$

The conjecture is stated as a refinement of the corresponding divisibility corollary, and the source leaves its full generality unresolved.