Fu–Peng–Zhang's multiplicability conjecture for finitary upho posets

Let $P$ be an upho poset. It is multiplicable if it admits a left-cancellative, invertible-free monoid structure $(P,*,e)$ whose left-divisibility order is the given order, namely

$$a\leq b\iff \exists c,\ a*c=b.$$

Here $P$ is finitary when its intervals are finite.

Fu–Peng–Zhang's multiplicability conjecture. Every finitary upho poset, in particular every finite-type $\mathbb{N}$-graded upho poset, is multiplicable.

The paper disproves this conjecture using a finitary upho poset constructed from walks in the Petersen graph. Thus finitary and finite-type $\mathbb{N}$-graded hypotheses do not suffice for multiplicability.