Gal's nonnegativity conjecture for the second gamma-number of flag spheres

Let $\Delta$ be a flag simplicial $d$-sphere. Its second gamma-number $\gamma_2(\Delta)$ is defined by

$$\gamma_2(\Delta)=f_1(\Delta)-(2d-1)f_0(\Delta)+(d-1)(2d+2),$$

where $f_0(\Delta)$ and $f_1(\Delta)$ are the numbers of vertices and edges, respectively. Gal's conjecture.

$$\gamma_2(\Delta)\geq 0.$$

This is known in dimension three and for several other classes of simplicial spheres. The paper also establishes the inequality for flag normal $d$-pseudomanifolds with at most $2d+7$ vertices, but the general case remains open.