A q-rious positivity

The $q$-binomial coefficients $\text{qbinom}nm={\prod }_{i=1}^m(1-q^{n-m+i})\mathrm{/}(1-q^i)$, for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the many combinatorial interpretations of $\text{qbinom}nm$. In this note we conjecture an arithmetically motivated generalisation of the non-negativity property for products of ratios of $q$-factorials that happen to be polynomials.