The Billera–Lee upper-bound conjecture for triangulated spheres

Let $S$ be a triangulated $d$-sphere, and let $T$ be the Billera–Lee $d$-sphere with $f$-vector $f(S)$.

Billera–Lee upper-bound conjecture. For every integer $i$ with $0\leq i\leq d$,

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

This conjecture proposes that Billera–Lee spheres maximize the entire $\tau$-vector among triangulated spheres with a fixed $f$-vector. The source does not state a resolution.