Generalized alternating binomial-sum divisibility conjecture

Let $n_1,\ldots,n_m$ be positive integers, and set $n_{m+1}=n_1$. The sum

$$\sum_{k=-n_1}^{n_1}(-1)^k\prod_{i=1}^m {n_i+n_{i+1}+1\choose n_i+k}{n_i+n_{i+1}+1\choose n_i+k+1}$$

Generalized divisibility conjecture. is congruent to zero modulo

$${n_1+n_m+1\choose n_1}\prod_{i=1}^{m-1}(n_i+n_{i+1}+1).$$

This is presented as a further generalization of the paper's preceding theorems; no resolution is given in the supplied text.