Athanasia's gamma-vector monotonicity conjecture for flag homology subdivisions

Let $\Delta$ be a flag homology sphere, and let $\Delta'$ be a flag vertex-induced homology subdivision of $\Delta$. The $\gamma$-vectors $\gamma(\Delta)$ and $\gamma(\Delta')$ are defined from the symmetric $h$-polynomials of these homology spheres by

$$h(\Delta,x)=\sum_i\gamma_i(\Delta)x^i(1+x)^{d-2i}.$$

Athanasia's conjecture. One has

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

coordinatewise. The conjecture asks for a monotonicity principle extending the behavior of $\gamma$-vectors under suitable subdivisions. It is known whenever $\Delta$ has dimension at most four, but remains open in general.