The antichain obstruction conjecture for projectively unique order polytopes

Let $\operatorname{\mathscr{P}}$ be a finite poset. Its order polytope $\operatorname{\mathcal{O}}(\operatorname{\mathscr{P}})$ is the polytope associated with $\operatorname{\mathscr{P}}$ by the order-polytope construction. A poset has an antichain of size $3$ if it contains three pairwise incomparable elements.

Antichain obstruction conjecture. If $\operatorname{\mathcal{O}}(\operatorname{\mathscr{P}})$ is projectively unique, then $\operatorname{\mathscr{P}}$ has no antichain of size $3$.

This conjecture asserts that the absence of a three-element antichain is necessary for projective uniqueness of an order polytope. The paper presents it as a proposed necessary condition; its general validity remains open.