Guo–Jouhet–Zeng divisibility conjectures for alternating binomial products

Let $r,s,t$ and $n$ be positive integers, and set

$$S_{r,s,t}(n)=\sum_{k=-n}^{n}(-1)^k\binom{6n}{3n+k}^r\binom{4n}{2n+k}^s\binom{2n}{n+k}^t.$$

Also set

$$T_{r,s,t}(n)=\sum_{k=-n}^{n}(-1)^k\binom{8n}{4n+k}^r\binom{4n}{2n+k}^s\binom{2n}{n+k}^t.$$

Guo–Jouhet–Zeng's divisibility conjecture. For all positive integers $r,s,t,n$,

$$S_{r,s,t}(n)\equiv 0\pmod{2\binom{6n}{n}},$$ $$S_{r,s,t}(n)\equiv 0\pmod{6\binom{6n}{3n}}.$$

Furthermore, if $(r,s,t)\ne(1,1,1)$, then

$$T_{r,s,t}(n)\equiv 0\pmod{2\binom{8n}{3n}}.$$

These claims refine divisibility phenomena for alternating products of binomial coefficients. The source presents them as computer-experiment conjectures and supplies no resolution evidence in the provided text.