Multivariate positivity conjecture for alternating q-binomial sums

Let $n_1,\ldots,n_m$ be positive integers, set $n_{m+1}=n_1$, and let $0\leqslant j\leqslant 2m-1$. The expression

$${n_1+n_m+1\brack n_1}^{-1}\prod_{i=1}^{m-1}\frac{1}{[n_i+n_{i+1}+1]} \sum_{k=-n_1}^{n_1}(-1)^k q^{jk^2+{k\choose 2}} \prod_{i=1}^m {n_i+n_{i+1}+1\brack n_i+k}{n_i+n_{i+1}+1\brack n_i+k+1}$$

Multivariate positivity conjecture. is a polynomial in $q$ with nonnegative integer coefficients. This is stated as a generalization of the q-Narayana positivity conjecture; no resolution is given in the supplied text.