A cyclic q-Narayana-type positivity conjecture

Let $m$ be a positive integer, let $n_1,\ldots,n_m$ be positive integers, and set $n_{m+1}=n_1$. Let $j$ be an integer and let $r$ be a non-negative integer. The cyclic q-Narayana-type conjecture. The expression

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

is a Laurent polynomial in $q$, and it has non-negative integer coefficients when $0\leqslant j\leqslant 2m$. When all $n_i$ are equal, the claim reduces to the preceding q-Narayana corollary. The paper presents it as its final conjecture and leaves it open.