The gamma2 conjecture for flag simplicial spheres

Let $d$ be a positive integer, let $\Delta$ be a flag simplicial $(d-1)$-sphere, and define

$$\gamma_2(\Delta)=f_1(\Delta)-(2d-3)f_0(\Delta)+2d(d-2).$$

The gamma2 conjecture.

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

This is a special case of the gamma-conjecture for flag spheres. It is known for flag simplicial 3-spheres by the Davis–Okun theorem and for several other classes, but remains open for general flag spheres.