Gatzweiler–Krattenthaler conjecture on positivity of Stanton fractions

Let $m,n$ be positive integers with $m\geq n$, let $\mathbf{a}=(a_1,\ldots,a_n)$ be a sequence of non-negative integers, and consider

$$\frac{\prod_{i=1}^n(1-q^{m+i})^{a_i}}{\prod_{i=1}^n(1-q^i)^{a_i}}.$$

Gatzweiler–Krattenthaler conjecture. If this fraction is a polynomial in $q$, then it has non-negative coefficients. This is presented as a strengthening of Stanton's conjecture, removing both the assumed polynomiality for every non-negative $m$ and the symmetry condition on $\mathbf{a}$. The supplied text does not report a proof or disproof of this strengthened claim.