Submultiplicativity conjecture for the flag $f$-vector coefficients of CW-posets

Let $P$ be a Cohen–Macaulay CW-poset or a Gorenstein* poset of rank $n$, let $\mathcal A_n$ be the indexing family for its flag $f$-vector coefficients, and let $\alpha_S(P)$ denote the coefficient indexed by $S\in\mathcal A_n$. For a partition $S=T_1\cup T_2$ of $S$, the submultiplicativity conjecture.

$$\alpha_S(P)\leq \alpha_{T_1}(P)\alpha_{T_2}(P).$$

This conjecture generalizes the inequalities established in the proof of Theorem 6.5. It concerns proposed multiplicative upper bounds for flag $f$-vector data of Cohen–Macaulay CW-posets and Gorenstein* posets; no resolution is supplied in the source.