Rank-generating realization conjecture for finite-type graded upho posets

Let $P$ be a finite-type $\mathbb{N}$-graded upho poset, and let $P_n$ denote the set of elements of rank $n$. A monoid is LCH if it is left-cancellative and homogeneous; for an LCH monoid $M$, let $M_n$ denote the set of elements of length $n$.

Rank-generating realization conjecture. There exists a finitely generated LCH monoid $M$ such that

$$|P_n|=|M_n|\quad\text{for all }n\in\mathbb{N}.$$

This weaker form of multiplicability remains plausible after the Petersen counterexample: the poset itself need not be multiplicable, while its rank-generating function may still be realized by a finitely generated LCH monoid. The paper gives such a realization for the Petersen example but does not establish the general statement.