A q-super Catalan Laurent-polynomial conjecture

Let $m$, $n$, $s$ and $t$ be positive integers. Let $r$ be a non-negative integer with $r+s+t\equiv 1\pmod{2}$, and let $j$ be an integer. Let $B_{m,k}(q)$ and $B_{n,k}(q)$ denote the $q$-binomial expressions defined earlier in the paper. The q-super Catalan conjecture. The expression

$$\frac{[m+n]![m]![n]!}{[2m]![2n]!} \sum_{k=1}^{m}(1+q^k)[k]^{r}q^{jk^2-(r+s+t+1)k/2} B_{m,k}^s(q) B_{n,k}^t(q)$$

is a Laurent polynomial in $q$, and it has non-negative integer coefficients when $0\leqslant j\leqslant s+t$. This refines the preceding result on $q$-super Catalan numbers and is a $q$-analogue of a conjecture of Guo and Zeng; its status is presented in the paper as an open problem.