Seo–Yee nonnegativity conjecture for a nonmodular infinite product

Throughout, use the $q$-series notation

$$(A;q)_\infty:=\prod_{k\ge 0}(1-Aq^k),$$

and $(A,B,\ldots,C;q)_\infty:=(A;q)_\infty(B;q)_\infty\cdots(C;q)_\infty$. Seo–Yee's nonnegativity conjecture. The series expansion of

$$\frac{1}{(q,-q^3;q^4)_\infty}$$

has nonnegative coefficients. Seo and Yee showed that this nonnegativity conjecture is equivalent to an earlier conjecture of Coll, A. Mayers and N. Mayers concerning the index of seaweed algebras and integer partitions; the claim concerns positivity properties of a generally nonmodular infinite product and is the focus of the paper's asymptotic and circle-method approach.