Nevo's gamma2 equality conjecture for flag spheres

Let $d\geq 4$ be an integer and let $\Delta$ be a flag simplicial $(d-1)$-sphere. Define $\gamma_2(\Delta)=f_1-(2d-3)f_0+2d(d-2)$. An edge contraction replaces an edge $e$ by the contracted complex $\Delta/e$, and $\operatorname{lk}_{\Delta}(e)$ denotes its link. Nevo's gamma2 equality conjecture. The following are equivalent: $\gamma_2(\Delta)=0$; and there is a sequence of edge contractions from $\Delta$ to the boundary of the $d$-dimensional cross polytope, namely the octahedral $(d-1)$-sphere, such that every complex in the sequence is a flag sphere. Moreover, the link of every contracted edge is the octahedral $(d-3)$-sphere. The implication from the contraction characterization to vanishing $\gamma_2$ is described as easy, but the supplied text gives no resolution of the full equivalence.