Maximality of lcm lattices of maximal monomial ideals in Betti strata

Let $X$ be a regular cell complex, and let $M\in CM_*(X)$ be a maximal monomial ideal with lcm lattice $P$. Let $\mathcal{L}(n)$ be the relevant poset of lcm lattices, ordered by $Q>P$. The total Betti numbers of a lattice are the total Betti numbers of its minimal free resolution.

Maximality conjecture. If $Q>P$ in $\mathcal{L}(n)$, then the minimal resolution of $Q$ has total Betti numbers greater than those of $P$. Equivalently, $P$ is maximal in its Betti stratum.

The conjecture generalizes the preceding argument for tree-supported resolutions and would provide an alternate description of the elements on the boundary of these Betti strata. The supplied text gives no resolution status.