Hibi–Li's f-vector conjecture for order and chain polytopes

Let $P$ be a finite poset with $|P|=d>1$. Write $f_i(Q)$ for the number of $i$-dimensional faces of a polytope $Q$, and let ${\mathcal O}(P)$ and ${\mathcal C}(P)$ denote the order polytope and chain polytope of $P$, respectively. Hibi–Li's conjecture. For every $i$ with $1\leq i\leq d-1$,

$$f_i({\mathcal O}(P))\leq f_i({\mathcal C}(P)).$$

Moreover, if equality holds for some $i$ with $1\leq i\leq d-1$, then ${\mathcal O}(P)$ and ${\mathcal C}(P)$ are unimodularly equivalent. Hibi and Li proposed this as a comparison of the face numbers of order and chain polytopes; the paper's abstract proves the relevant inequality for triangular $2$-faces when $P$ is maximal ranked, but the full conjecture is not resolved here.