The pop-stack-sorting operator on Tamari lattices

Motivated by the pop-stack-sorting map on symmetric groups, Defant defined an operator $\mathsf{Pop}_M : M \to M$ for each complete meet-semilattice $M$ $$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}).$$ This paper concerns dynamics of $\mathsf{Pop}_{\mathrm{Tam}_n}$, where $\mathrm{Tam}_n$ is $n$-th Tamari lattice. We say element $x\in \mathrm{Tam}_n$ $t$-$\mathsf{Pop}$-sortable if $\mathsf{Pop}_M^t (x)$ minimal and we let $h_t(n)$ denote number elements in $\mathrm{Tam}_n$. find explicit formula generating function $\sum_{n\ge 1}h_t(n)z^n$ verify Defant's conjecture that it rational. furthermore prove size image $\mathsf{Pop}_{\mathrm{Tam}_n}$ Motzkin $M_n$, settling a Williams.