Athanasia's local gamma-vector nonnegativity conjecture for flag vertex-induced subdivisions

Let $V$ be a $d$-element set, and let $\Gamma$ be a flag vertex-induced homology subdivision of the simplex $2^V$. The local $h$-polynomial has a unique expansion

$$\ell_V(\Gamma,x)=\sum_{i=0}^{\lfloor d/2\rfloor}\xi_i x^i(1+x)^{d-2i},$$

and $\xi_V(\Gamma)=(\xi_0,\dots,\xi_{\lfloor d/2\rfloor})$ is the local $\gamma$-vector of $\Gamma$. Athanasia's conjecture. The local $\gamma$-vector $\xi_V(\Gamma)$ has nonnegative coordinates for every flag vertex-induced homology subdivision $\Gamma$ of the simplex $2^V$. This conjecture is motivated by the local decomposition of $\gamma$-vectors for homology subdivisions and is open in general; the source records partial results in low dimensions and for subdivisions obtained by successive stellar subdivisions on edges.