Guo–Zeng's three-factor ballot-number congruences

Let $A_{n,k}$ be the ballot numbers

$$A_{n,k}={2n\choose n-k}-{2n\choose n-k-1}.$$

For all $n,r,s,t\in\mathbb{Z}^+$ with $r+s+t\equiv1\pmod2$, and with $\varepsilon=\pm1$, Guo–Zeng's three-factor ballot-number conjecture. The following congruences should hold:

$$(4n+1)\sum_{k=0}^n\varepsilon^kA_{3n,k}^rA_{2n,k}^sA_{n,k}^t\equiv0\pmod{\frac{1}{6n+1}{6n+1\choose n}},$$ $$(4n+1)\sum_{k=0}^n\varepsilon^kA_{3n,k}^rA_{2n,k}^sA_{n,k}^t\equiv0\pmod{\frac{1}{6n+1}{6n+1\choose 3n}},$$ $$\sum_{k=0}^n\varepsilon^kA_{4n,k}^rA_{2n,k}^sA_{n,k}^t\equiv0\pmod{\frac{1}{8n+1}{8n+1\choose 3n}},$$

and

$$(6n+1)\sum_{k=0}^n\varepsilon^kA_{4n,k}^rA_{3n,k}^sA_{2n,k}^t\equiv0\pmod{\frac{1}{8n+1}{8n+1\choose 3n}}.$$

These are proposed extensions of the preceding proved congruences for products of binomial coefficients.