Nevo–Petersen conjecture on gamma-vectors of flag simplicial spheres

Let $\Delta$ be a flag simplicial sphere of dimension $d-1$, and let

$$\gamma(\Delta):=(\gamma_0(\Delta),\ldots,\gamma_{\lfloor d/2\rfloor}(\Delta))$$

be the coefficients in its gamma expansion. Nevo–Petersen conjecture. There exists a simplicial complex $\Gamma$ such that $\gamma(\Delta)=f(\Gamma)$.

This strengthens Gal’s conjecture by requiring the gamma-vector to satisfy the Kruskal–Katona inequalities. The source does not state a resolution.