The Hibi–Li (2016) conjecture is fully resolved, with miscellanea on marked chain-order polytopes

We prove the equality part of the Hibi–Li (2016) conjecture, its last open part: for every n-element poset P and every 2 ≤ k ≤ n−1, if the chain polytope 𝒞(P) and the order polytope 𝒪(P) have the same number of k-dimensional faces, then they are unimodularly equivalent. Its inequality part, that 𝒞(P) has at least as many k-dimensional faces as 𝒪(P), was first proved by Fourier; for edges the equality part fails, since the two polytopes always have the same number of edges. So the conjecture is now fully resolved. Both parts follow from two theorems. First, the face numbers of a bounded marked chain-order polytope weakly increase when its chain part grows; the partition is arbitrary, and the marking may be real and degenerate. Second, for an ordinary n-element poset whose order and chain polytopes are not unimodularly equivalent, we prove the sharp inequalities f_k(𝒞(P))−f_k(𝒪(P)) ≥ (n − 3 choose k − 2) for 2 ≤ k ≤ n−1. We characterize simultaneous equality and all tight indices, and obtain a sharper bound with prescribed facet difference within the two-face equality class. For bounded nested marked partitions, equality of facet numbers is equivalent to affine equivalence; integral markings give lattice equivalence. We prove that regular marked posets have irredundant canonical facet descriptions for every partition, including when maximal elements are unmarked. In that unbounded setting a switch weakly increases or weakly decreases every face number according to whether it has an upper endpoint. Nonzero changes have terminal support, and their quotients by (1 + t)², respectively by 1 + t, have coefficients of the sign of the regime. A connected example shows that opposite regimes can cancel along a nested sequence. For bounded marked posets and several pairwise incomparable switched order elements, the alternating sum of face polynomials is a higher-order subdivision defect. It obeys a sign rule under mutual transversality, and for layered posets whose distinct lower and upper neighbourhoods form Berge-acyclic families; otherwise the sign rule can fail, and the defect can have coefficients of both signs. When all these neighbourhoods have two elements, the defect is, up to sign, a power of 2 + t times the chromatic polynomials of the lower and the upper neighbourhood graph, evaluated at −t; the sign rule then holds exactly when both graphs are forests. For neighbourhoods of any size, the defect of a layered poset is, up to sign and a power of 2 + t, a product of two alternating sums of chromatic polynomials of hypergraphs at −t; for neighbourhoods with three or more elements, Berge-acyclicity is not necessary for the sign rule. The arguments combine fixed-height lifts and clipped markings with general transverse-subdivision results proved in the appendices.