Kalai's generalized upper bound conjecture for shellable CW spheres

Let the Generalized Upper Bound Theorem mean that, for a simplicial $d$-polytope $P$ and the cyclic $d$-polytope $C(d,n)$, the inequality $f_{d-1}(P) \geq f_{d-1}(C(d,n))$ implies $f_k(P) \geq f_k(C(d,n))$ for $k=0,\ldots,d-1$. A shellable, strongly regular CW sphere is a shellable, strongly regular CW complex homeomorphic to a sphere. Kalai's conjecture. The Generalized Upper Bound Theorem applies to arbitrary shellable, strongly regular CW spheres. The source places this conjecture between the corresponding conjectures for arbitrary polytopes and arbitrary Eulerian lattices; it remains open according to the source.