The rational Catalan area-codinv conjecture for coprime Dyck paths

Let $m$ and $n$ be coprime positive integers. An $(m,n)$-Dyck path is a lattice path from $(0,0)$ to $(m,n)$ staying on the prescribed side of the main diagonal; let $\operatorname{area}(D)$ be the number of lattice squares between $D$ and the main diagonal, and let $\operatorname{codinv}(D)$ be the Dyck path statistic given by a simple geometric construction. Define $[a]_q=1+q+\cdots+q^{a-1}$ and $\left[m+n \atop m,n \right]_q$ as the relevant $q$-multinomial coefficient. The rational Catalan area-codinv conjecture.

$$\sum_{D\in \mathcal D_{m,n}}q^{\operatorname{area}(D)+\operatorname{codinv}(D)}=\operatorname{Cat}_{m,n}(q)=\frac{1}{[m+n]_q}\left[m+n \atop m,n \right]_q.$$

This conjecture seeks a combinatorial interpretation of the rational Catalan polynomial and arose from the rational Shuffle conjecture; it was also formulated in related work of Armstrong and collaborators. The source provides no resolution status.