Marked chain-order polytope face-number monotonicity conjecture

Let $(P,A,\mathbf{\lambda})$ be a regular marked poset, and let $\operatorname*{St}(P)$ denote the set of star elements. An admissible decomposition of $P\setminus A$ is a pair $(U_1,U_2)$ with $U_1\cup U_2=P\setminus A$, $U_1\cap U_2=\emptyset$, and no element of $U_1$ covered by an element of $U_2$; write $\mathcal{CO}_{U_1,U_2}(\mathbf{\lambda})$ for the associated marked chain-order polytope. Let $(U_1,U_2)$ and $(V_1,V_2)$ be admissible decompositions such that

$$U_1\cap\operatorname*{St}(P)=(V_1\cap\operatorname*{St}(P))\cup\{p\}$$

for $p\in\operatorname*{St}(P)$. Marked chain-order polytope face-number monotonicity conjecture. For every $0\leq i\leq |P\setminus A|$, the number of $i$-dimensional faces of $\mathcal{CO}_{V_1,V_2}(\mathbf{\lambda})$ is greater than or equal to the number of $i$-dimensional faces of $\mathcal{CO}_{U_1,U_2}(\mathbf{\lambda})$. This generalizes the Stanley–Hibi–Li conjecture and related conjectures for marked chain and marked order polytopes; the facet case follows from the paper's preceding theorem, while the asserted inequalities in all dimensions remain open.