Refined Stanley–Hibi–Li conjecture for marked chain-order polytopes

Let $(P,A,\mathbf{\lambda})$ be a marked poset, and let $(U_1,U_2)$ and $(V_1,V_2)$ be different admissible decompositions of $P\setminus A$, meaning that each pair partitions $P\setminus A$ and has no element of its first part covered by an element of its second part. Write $\mathcal{CO}_{U_1,U_2}(\mathbf{\lambda})$ for the corresponding marked chain-order polytope. Refined Stanley–Hibi–Li conjecture. If $U_1\subset V_1$, then for every $i=0,1,\ldots,|P\setminus A|$,

$$f_i(\mathcal{CO}_{V_1,V_2}(\mathbf{\lambda}))\leq f_i(\mathcal{CO}_{U_1,U_2}(\mathbf{\lambda})).$$

This refines the comparison between marked chain and marked order polytopes by comparing all admissible intermediate decompositions; the preceding theorem gives a strict facet decrease in the relevant star-element step, but the full face-vector inequalities are open.