Nevo–Lutz's gamma-two rigidity conjecture for flag spheres

Let $d\geq 4$ be an integer and let $\Delta$ be a flag simplicial $(d-1)$-sphere. Nevo–Lutz's gamma-two conjecture. The following are equivalent:

1. $\gamma_2(\Delta)=0$.
2. There is a sequence of edge contractions from $\Delta$ to the boundary of the $d$-cross-polytope such that all complexes in the sequence are flag spheres, and the link of each contracted edge is the octahedral $(d-3)$-sphere.

This is a proposed characterization of flag spheres with vanishing second $\gamma$-number; the source says it is still open.