The unranked graphic order polytope conjecture

Let $\operatorname{\mathscr{P}}$ be a finite poset, not necessarily ranked, with no antichain of size $3$. Its order polytope $\operatorname{\mathcal{O}}(\operatorname{\mathscr{P}})$ is the polytope associated with $\operatorname{\mathscr{P}}$ by the order-polytope construction.

Unranked graphic order polytope conjecture. The order polytope $\operatorname{\mathcal{O}}(\operatorname{\mathscr{P}})$ is graphic.

This conjecture proposes dropping the ranked hypothesis from the preceding theorem characterizing a class of graphic order polytopes. The paper notes that rankedness is not necessary and gives non-ranked graphic examples, but does not establish the conjecture in general.