Factors of sums involving $q$-binomial coefficients and powers of $q$-integers

We show that, for all positive integers $n_1,\dots ,n_m$, $n_{m+1}=n_1$, and any non-negative integers $j$ and $r$ with $j⩽m$, the expression $$\frac{1}{[n_1]}{\left[\genfrac{}{}{0px}{}{n_1+n_m}{n_1}\right]}^{-1}\sum _{k=1}^{n_1}[2k][k{]}^{2r}{q}^{j{k}^2-(r+1)k}\prod _{i=1}^m\left[\genfrac{}{}{0px}{}{n_i+n_{i+1}}{n_i+k}\right]$$ is a Laurent polynomial in $q$ with integer cofficients, where $[n]=1+q+\cdots +q^{n-1}$ and $\left[\genfrac{}{}{0px}{}{n}{k}\right]={\prod }_{i=1}^k(1-q^{n-i+1})\mathrm{/}(1-q^i)$. This gives a $q$-analogue of a divisibility result on the Catalan triangle obtained by the first author and Zeng, and also confirms a conjecture of the first author and Zeng. We further propose several related conjectures.