Nonnegative coefficients for polynomial fake Gaussian sequences

Prove that for given positive integers m and n and every symmetric sequence of non-negative integers \((a_1,a_2,\ldots,a_n)\), if the expression \(\prod_{i=1}^{n}\frac{(1-q^{m+i})^{a_i}}{(1-q^i)^{a_i}}\) is a polynomial in q, then it has non-negative coefficients.

Background

This conjecture extends the q-binomial positivity conjecture to D. Stanton’s fake Gaussian sequences. The sequence of exponents is required to be symmetric and consist of non-negative integers, while the polynomiality condition is imposed only for the specified value of m.

The authors explain that Stanton’s original conjecture requires the expression to be a polynomial for every positive integer m, whereas Conjecture 2 drops that stronger polynomiality requirement. They report computational evidence for randomly generated symmetric sequences and verify several structured special cases, but do not prove the general assertion.