Strong monotonicity conjecture for flag vertex-induced homology subdivisions

Let $\Delta$ be a flag homology sphere, and let $\Delta'$ be a flag vertex-induced homology subdivision of $\Delta$. Let $\gamma(\Delta)$ and $\gamma(\Delta')$ denote their $\gamma$-vectors. Strong monotonicity conjecture. For every flag homology sphere $\Delta$ and every flag vertex-induced homology subdivision $\Delta'$ of $\Delta$, we have

$$\gamma(\Delta')\geq\gamma(\Delta).$$

This is presented as a stronger version of the Postnikov–Reiner–Williams monotonicity conjecture. The paper proves the assertion in dimensions 3 and 4; the unrestricted dimensional statement remains open.