A positivity conjecture for a quotient of q-binomial coefficients

We conjecture that, if the quotient of two <i>q</i>-binomial coefficients with the same top argument is a polynomial, then it has non-negative coefficients. We summarise what is known about the conjecture and prove it in two non-trivial cases. Moreover, we move ahead to extend our conjecture to D. Stanton's fake Gaussian sequences. As a corollary of one of our results we obtain that a polynomial that is conjectured to be a cyclic sieving polynomial for Kreweras words [S. Hopkins and M. Rubey, <i>Selecta Math. (N.S.)</i> <b>28</b> (2022), Paper No. 10] is indeed a polynomial with non-negative integer coefficients.