Stanley–Hibi–Li conjecture for marked chain and order polytopes

Let $(P,A,\mathbf{\lambda})$ be a marked poset. For an $N$-dimensional polytope $Q$, let $f_i(Q)$ denote the number of its $i$-dimensional faces, and write $\mathcal{C}_{P,A}(\mathbf{\lambda})$ and $\mathcal{O}_{P,A}(\mathbf{\lambda})$ for the marked chain and marked order polytopes. Stanley–Hibi–Li conjecture. For all $i=0,1,\ldots,|P\setminus A|$,

$$f_i(\mathcal{C}_{P,A}(\mathbf{\lambda}))\geq f_i(\mathcal{O}_{P,A}(\mathbf{\lambda})).$$

The original conjecture concerned unmarked chain and order polytopes; the facet case is known for marked chain and marked order polytopes, while the zero-dimensional marked case remains open, and the full marked statement is therefore unresolved.