Centrally symmetric cyclic upper bound conjecture for odd-dimensional spheres

Let $d$ be even. A nearly neighborly centrally symmetric $d$-sphere is a centrally symmetric $d$-dimensional triangulated sphere whose faces satisfy the nearly neighborly condition, and a vertex-transitive cyclic group action is a cyclic group action transitive on its vertices. Let $C_d^{\Delta}$ denote the $d$-dimensional crosspolytope and let $\partial C_d^{\Delta}$ be its boundary complex.

Centrally symmetric cyclic upper bound conjecture. If $d$ is even, then the boundary complex $\partial C_d^{\Delta}$ of the $d$-dimensional crosspolytope on $n=2d$ vertices is the only nearly neighborly centrally symmetric $d$-sphere with a vertex-transitive cyclic group action.

This is a uniqueness assertion for the extremal case of the centrally symmetric cyclic upper bound problem. The surrounding discussion notes that the broader upper bound conjecture is intended for odd-dimensional spheres and that the corresponding existence statement is trivial in dimension one; the resolution status of this uniqueness claim is not specified in the source.