A q-rious positivity

The q-binomial coefficients ${\genfrac{}{]}{0pt}{}{n}{m}= \prod_{i=1}^m (1-q^{n-m+i})/(1-q^i)}$ , for integers 0??? m??? 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 ${\genfrac{}{]}{0pt}{}{n}{m}}$ . 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.