Rationality conjecture for sortable elements of m-Tamari lattices

For integers $m,t\geq 1$, let $h_t(m,n)$ denote the number of $t$-$\operatorname{\mathsf{Pop}}$-sortable $m$-ballot paths in the $m$-Tamari lattice $\operatorname{Tam}_n(m)$. Rationality conjecture. For fixed $m$ and $t$, the generating function

$$\sum_{n\geq 1}h_t(m,n)z^n$$

is rational. This conjecture extends the enumerative results for $1$- and $2$-$\operatorname{\mathsf{Pop}}$-sortable paths; rationality is asserted for every fixed pair $m,t$, but no proof or resolution is supplied here.