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

Let O(P) and 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 O(P) is less than or equal to that of C(P), with equality holding if and only if P does not contain an X-poset as a subposet.