Athanasiadis's gamma-positivity conjecture for flag simplex triangulations

Let $V$ be an $n$-element set, let $2^V$ denote the simplex on $V$, and let $\Gamma$ be a flag triangulation of $2^V$. Let $ell_V(\Gamma,x)$ denote its local $h$-polynomial. Athanasiadis's conjecture. The polynomial $ell_V(\Gamma,x)$ is $\gamma$-positive for every flag triangulation $\Gamma$ of the simplex $2^V$. This is an analogue of Gal's conjecture for local $h$-polynomials and is stated more generally in the cited work for a class of topological simplicial subdivisions; the source gives no resolution.