The positive rationality conjecture for normalized skew $q$-hook-content formulas

Let $\mu\subseteq\lambda$ be partitions, let $\ell$ be the length of $\lambda$, and fix an integer $n\geq\ell$. Let $H_{\lambda/\mu}(n;q)$ be the skew $q$-hook-content quantity and let $[\lvert\lambda/\mu\rvert]_q!$ denote the corresponding $q$-factorial. The normalized positive rationality conjecture. There exist $a(q),b(q)\in\mathbb{Z}_{\geq0}[q]$ such that

$$\frac{H_{\lambda/\mu}(n;q)}{[\lvert\lambda/\mu\rvert]_q!}=\frac{a(q)}{b(q)},\qquad a(-1)\in\mathbb{Z}_{\geq0}.$$

The conjecture is based on experimental evidence and concerns the normalized expression even when it is not polynomial. The supplied text gives no proof or resolution.