Nevo's balanced-complex strengthening of Gal's conjecture

Let $\Delta$ be a flag homology sphere, and let $\gamma(\Delta)$ denote its $\gamma$-vector. Nevo's strengthening of Gal's conjecture. The vector $\gamma(\Delta)$ is the $f$-vector of some balanced simplicial complex. This strengthens Gal's nonnegativity conjecture and was proposed as a further strengthening in the cited work. The statement remains open in general.