The Billera–Lee upper-bound conjecture for spheres

Let $S$ be a $d$-sphere with the $f$-vector of some $(d+1)$-polytope, and let $T$ be the Billera–Lee $d$-sphere with $f$-vector $f(S)$.

Billera–Lee upper-bound conjecture. The inequality

$$\tau_i(S) \leq \tau_i(T)$$

holds for all $i$.

This extends the known result for polytopal spheres to non-polytopal spheres having the $f$-vector of a polytope, and also removes the characteristic-zero restriction; the source does not state a resolution.