Redundant symmetry-rigidity conjecture for symmetric simplicial circuits

Let $({\cal S},*)$ be a ${\mathbb Z}_2$-symmetric simplicial $k$-circuit with $k\geq 3$, ${\cal S}\neq {\rm?}$; here ${\cal B}_k$ denotes the excluded complex from the paper. Assume that $G({\cal S})$ is $(k+1)$-connected. For a point group $\Gamma$ in $\mathbb R^{k+1}$ of order two and an edge $e\in E({\cal S})$, let $e^*$ be its image under the involution, and let $G({\cal S})-e-e^*$ denote deletion of both edges.

Redundant symmetry-rigidity conjecture. For every such $\Gamma$ and $e$, $(G({\cal S})-e-e^*,*)$ is $\Gamma$-rigid in $\mathbb R^{k+1}$.

The conjecture would imply that equality in the symmetric lower bound theorem occurs only for symmetrically stacked spheres and would establish a special case of the lower bound conjecture cited by the authors. Its status is not resolved in the supplied text.