Hibi–Li conjecture on the face numbers of order and chain polytopes

Let $P$ be a finite poset with $|P|=d>1$. For a polytope ${\mathcal P}$, let $f_i({\mathcal P})$ denote the number of its $i$-dimensional faces. Let ${\mathcal O}(P)$ and ${\mathcal C}(P)$ denote the order and chain polytopes of $P$, respectively. Two polytopes are unimodularly equivalent if one is the image of the other under an affine transformation whose linear part is unimodular.

Hibi–Li conjecture.

$$f_i({\mathcal O}(P))\leq f_i({\mathcal C}(P))\qquad\text{for all }1\leq i\leq d-1.$$

Moreover, if

$$f_i({\mathcal O}(P))=f_i({\mathcal C}(P))$$

for some $2\leq i\leq d-1$, then ${\mathcal O}(P)$ and ${\mathcal C}(P)$ are unimodularly equivalent.

The paper proves the case $i=2$, showing that the chain polytope has at least as many triangular faces as the order polytope and equally many square faces, with strict inequality except in the unimodularly equivalent case. The remaining cases of the conjecture are not resolved here.