The rectangular shuffle theorem for rectangular paths

Let $m,n\in\mathbb{N}$, let $d=\gcd(m,n)$, and let $p_{m,n}$ be the symmetric function associated with rectangular paths. For a rectangular path $\pi\in\mathsf{LRP}(m,n)$, let $\mathsf{dinv}(\pi)$ and $\mathsf{area}(\pi)$ denote its statistics, and let $x^\pi$ denote its monomial weight. Write $[r]_q$ for the $q$-integer. Rectangular paths conjecture. One has

$$\frac{[m]_q}{[d]_q}p_{m,n}=\sum_{\pi\in\mathsf{LRP}(m,n)}q^{\mathsf{dinv}(\pi)}t^{\mathsf{area}(\pi)}x^\pi.$$

The conjecture extends the rectangular shuffle theorem from rectangular Dyck paths to rectangular paths. The supplied text gives no resolution evidence for this statement, although it notes verification by computer in a bounded range and a proof in the special case of coprime side lengths.