The positive rationality conjecture for skew $q$-hook-length polynomials

Let $\mu\subseteq\lambda$ be partitions, and let $f^{\lambda/\mu}_q$ be the skew $q$-hook-length quantity. Assume it can be written as a rational function

$$f^{\lambda/\mu}_q=\frac{a(q)}{b(q)},$$

where $a(q),b(q)\in\mathbb{Z}_{\geq0}[q]$. The positive rationality conjecture. The numerator can be chosen so that

$$a(-1)\in\mathbb{Z}_{\geq0}.$$

This conjecture is motivated by experimental evidence after examples showing that $f^{\lambda/\mu}_q$ need not be a polynomial. It proposes positivity properties for a rational-function representation, but the supplied text gives no proof or resolution.