Multiplicability conjecture for finitary upho lattices

Let $P$ be an upho lattice, meaning an upho poset whose order is a lattice, and suppose that $P$ is finitary. It is multiplicable if it admits a left-cancellative, invertible-free monoid structure whose left-divisibility order is the given order.

Multiplicability conjecture for finitary upho lattices. Every finitary upho lattice, and in particular every finite-type $\mathbb{N}$-graded upho lattice, is multiplicable.

The Petersen counterexample is not a lattice, so it does not settle this restricted claim. The conjecture is connected with the classification of upho lattices by finite graded cores and would imply finiteness of the collection of upho lattices associated with a fixed core.