The infinitesimal rigidity conjecture for centrally symmetric homology spheres

Let $\Delta$ be a centrally symmetric simplicial complex of dimension $d-1\geq 3$. Assume that $\Delta$ is a homology sphere, a connected homology manifold, or a normal pseudomanifold. Then there exists a map $\mathbf p:V(\Delta)\to \mathbb R^d$ such that $(\Delta,\mathbf p)$ is a centrally symmetric framework that is infinitesimally rigid in $\mathbb R^d$. The infinitesimal rigidity conjecture. Such a realization should exist for every complex in the stated classes. This conjecture would imply the inequality part of the cs lower bound conjecture, but the source notes that links of centrally symmetric complexes are generally not centrally symmetric, obstructing standard inductive arguments.