Postnikov–Reiner–Williams monotonicity conjecture for subdivisions of flag homology spheres

Let $\Delta$ and $\Delta'$ be flag homology spheres, and suppose that $\Delta'$ geometrically subdivides $\Delta$. Let $\gamma(\Delta)$ and $\gamma(\Delta')$ denote their $\gamma$-vectors. Postnikov–Reiner–Williams conjecture. For all flag homology spheres $\Delta$ and $\Delta'$ for which $\Delta'$ geometrically subdivides $\Delta$, we have

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

This conjecture extends Gal's nonnegativity conjecture and is an analogue of monotonicity of the $h$-vector under suitable simplicial subdivisions. The paper proves it in dimensions 3 and 4 for flag vertex-induced homology subdivisions, while the general statement remains open.