Symmetric infinitesimal rigidity conjecture for centrally symmetric complexes

Let $\Delta$ be a $(d-1)$-dimensional centrally symmetric simplicial complex with an involution $\phi$, and let $G(\Delta)$ be its graph. A $d$-embedding is a map $\rho:V(\Delta)\to{\mathbb R}^d$ whose vertex positions give a $d$-dimensional framework; it respects symmetry when

$$\rho(\phi(v))=-\rho(v)$$

for every vertex $v$. The framework $(G(\Delta),\rho)$ is infinitesimally rigid when it has no nontrivial first-order edge-length-preserving deformations beyond Euclidean motions.

Symmetric infinitesimal rigidity conjecture. If $\Delta$ is a simplicial sphere, a connected simplicial manifold, or a normal pseudomanifold and $d-1\geq 3$, then there exists a symmetry-respecting $d$-embedding $\rho$ for which $(G(\Delta),\rho)$ is infinitesimally rigid.

If true, this would imply the inequality part of the centrally symmetric lower bound conjecture. The survey presents it as an open conjecture.