Guo–Zeng's mixed ballot-number congruences

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

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

For $r\in\mathbb{N}$, $s,t\in\mathbb{Z}^+$ with $r+s+t\equiv1\pmod2$, $n\in\mathbb{Z}^+$, and $\varepsilon=\pm1$, Guo–Zeng's adjacent-index congruence conjecture. The congruences

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

and

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

should hold for all such parameters, without the auxiliary gcd hypotheses appearing in the proved corollaries. The conjecture also includes the special case $s=t=1$.