Cyclic q-binomial Laurent-polynomial positivity conjecture

Let $n_1,\ldots,n_m,n_{m+1}=n_1$ be positive integers. For integers $j$ and non-negative integers $a,r$ with $a\leqslant n_1$, define the expression

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

Cyclic q-binomial conjecture. For every integer $j$, this expression is a Laurent polynomial in $q$, and it has non-negative integer coefficients when $0\leqslant j\leqslant m$. The conjecture is presented as a stronger version of an earlier theorem and as a generalization of a conjecture of Guo and Wang. It is listed among the paper's open problems.