Nevo--Petersen's flag-complex interpretation conjecture for gamma-vectors

Let $\Delta$ be a flag homology sphere. Write its $\gamma$-vector as $\gamma^\Delta=(1,\gamma_1,\gamma_2,\ldots)$. A simplicial complex is flag when its faces are exactly the cliques of its graph, equivalently when all its minimal non-faces have two elements. Nevo--Petersen's conjecture. The $\gamma$-vector of $\Delta$ is the $f$-vector of a flag simplicial complex. This strengthens Gal's nonnegativity conjecture, since $f$-vectors have nonnegative entries. The paper records partial results and derives bounds from this conjecture, but it remains open in general.