Kalai's $3^d$ conjecture asserts that every centrally symmetric $d$-polytope has at least $3^d$ non-empty faces. This is open in general, but has been proven for *simplicial polytopes*.

Question:Does the conjecture still hold for centrally symmetricsimplicial spheres? Or are counterexamples known?

Here, a simplicial sphere is a simplicial complex that is homeomorphic to a sphere, but is not neccessarily the boundary complex of a simplicial polytope. I am not sure how to best formalize "centrally symmetric" in this context, but one idea would be to require a fixed-point free involution on the sphere.