Loehr–Warrington nonstandard $q$-binomial formula conjecture

Fix $a,b\in\mathbb{N}$. Let $\mathrm{R}(a,b)$ be the set of partitions fitting in the $b\times a$ rectangle, let $\mathsf{ml}_{b,a}(\mu)$ be the minimum level on the frontier of $\mu$, and let $h^+_{b,a}(\mu)$ and $h^-_{b,a}(\mu)$ be the corresponding arm–leg statistics. Nonstandard $q$-binomial formula conjecture.

$$\genfrac{[}{]}{0pt}{}{a+b}{a,b}_q=\sum_{\mu\in\mathrm{R}(a,b)}q^{|\mu|+\mathsf{ml}_{b,a}(\mu)+h^+_{b,a}(\mu)}=\sum_{\mu\in\mathrm{R}(a,b)}q^{|\mu|+\mathsf{ml}_{b,a}(\mu)+h^-_{b,a}(\mu)}.$$

This gives a proposed combinatorial interpretation of the $q$-binomial coefficient without a coprimality assumption; the supplied text does not state its resolution.