Lutz–Nevo conjecture on flag PL-spheres with vanishing second gamma coefficient

Let $\Delta$ be a $(d-1)$-dimensional flag PL-sphere, meaning a piecewise-linear sphere whose simplicial complex is flag, and assume $d\geq 4$. Let $\gamma_2(\Delta)$ be its second gamma coefficient. Write $\Diamond_m$ for the cross-polytope boundary of dimension $m$; for an edge $F$ of a complex, let $\Delta/F$ denote its edge contraction and $\operatorname{lk}_{\Delta}(F)$ its link.

Lutz–Nevo conjecture. The following are equivalent:

1. $\gamma_2(\Delta)=0$.
2. There is a sequence of edge contractions

$$\Delta=\Delta_0\to\Delta_1=\Delta_0/F_1\to\cdots\to\Delta_{k-1}/F_k\cong\Diamond_{d-1},$$

where every $\Delta_i$ is a $(d-1)$-dimensional flag PL-sphere and

$$\operatorname{lk}_{\Delta_{i-1}}(F_i)\cong\Diamond_{d-3}$$

for every $1\leq i\leq k$.

The conjecture proposes a structural characterization of flag PL-spheres with $\gamma_2=0$. The paper studies the analogous structure for boundaries of symmetric edge polytopes; the general conjecture's status is not resolved by the supplied text and remains open.