Stanton's conjecture on symmetric fake Gaussian sequences

Let $\mathbf{a}=(a_1,\ldots,a_n)$ be a sequence of non-negative integers, and let $m$ be a non-negative integer. Consider

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

A sequence $\mathbf{a}$ is symmetric when its entries are symmetric in the sense intended for fake Gaussian sequences. Stanton's conjecture. If this fraction is a polynomial for all non-negative integers $m$ and $\mathbf{a}$ is symmetric, then it has non-negative integer coefficients for all $m$. The conjecture concerns positivity of a broad family of cyclotomic-type generating functions; the supplied text presents it as an earlier conjecture studied by Gatzweiler and Krattenthaler, without resolving it here.