Triangular faces of the order and chain polytope of a maximal ranked poset

Let $\mathscr{O}(P)$ and $\mathscr{C}(P)$ denote the order polytope and chain polytope, respectively, associated with a finite poset $P$. We prove the following result: if $P$ is a maximal ranked poset, then the number of triangular $2$-faces of $\mathscr{O}(P)$ is less than or equal to that of $\mathscr{C}(P)$, with equality holding if and only if $P$ does not contain an $X$-poset as a subposet.