Nonnegativity conjecture for local gamma-vectors of flag vertex-induced subdivisions

Let $V$ be the vertex set of a simplex $2^V$, and let $\Gamma$ be a flag vertex-induced homology subdivision of $2^V$. Let $\xi_V(\Gamma)$ denote the local $\gamma$-vector associated with the subdivision. Local gamma nonnegativity conjecture. For every flag vertex-induced homology subdivision $\Gamma$ of the simplex $2^V$ we have

$$\xi_V(\Gamma)\geq 0.$$

This is stated as the main conjecture of the paper and is motivated by the formula expressing the $\gamma$-vector of a subdivision of an Eulerian simplicial complex in terms of local $\gamma$-vectors. The supplied text does not state a resolution, so the conjecture remains open.