Guo–Zeng's general product ballot-number congruence

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

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

For all $n,r_1,\ldots,r_m\in\mathbb{Z}^+$ such that $r_1+\cdots+r_m\equiv1\pmod2$, and with $\varepsilon=\pm1$, Guo–Zeng's general ballot-product conjecture.

$$\sum_{k=0}^n\varepsilon^k\prod_{i=1}^mA_{n+i-1,k}^{r_i}\equiv0\pmod{\frac{1}{2n+1}{2n+1\choose n}}.$$

This is proposed as a challenging generalization of the preceding three-factor congruences and of the binomial-product congruences proved earlier in the paper.