The generalized lower bound conjecture for centrally symmetric simplicial spheres

Let $\Delta$ be a centrally symmetric simplicial complex of dimension $d-1\geq 3$, and let $g_2(\Delta)$ denote its second $g$-number. Let ${\mathcal C}^*_d$ be the $d$-dimensional cross-polytope; symmetric stacking means repeatedly attaching simplices along antipodal pairs of facets.

The centrally symmetric lower bound conjecture. If $\Delta$ is a simplicial sphere, a connected simplicial manifold, or a normal pseudomanifold, then

$$g_2(\Delta)\geq {d\choose 2}-d.$$

Moreover, equality holds if and only if $\Delta$ is the boundary complex of a centrally symmetric $d$-polytope obtained from ${\mathcal C}^*_d$ by symmetric stacking.

The survey states that this conjecture is wide open. It extends the known centrally symmetric lower bound result from simplicial polytopes to spheres, manifolds, and normal pseudomanifolds.