Kalai–Kleinschmidt–Lee conjecture on empty simplices of simplicial polytopes

Let $P$ be a simplicial $d$-polytope with prescribed $h$-vector $\underline{h}$, and let $P_{BL}(\underline{h})$ be the Billera–Lee simplicial $d$-polytope with $h$-vector $\underline{h}$. An empty simplex is a minimal non-face of $P$, and its dimension is one less than its cardinality.

Kalai–Kleinschmidt–Lee conjecture. For every $j$, the number of $j$-dimensional empty simplices of $P$ is at most the number of $j$-dimensional empty simplices of $P_{BL}(\underline{h})$.

Billera–Lee polytopes realize all admissible $h$-vectors and are expected to have extremal properties among simplicial polytopes with a fixed $h$-vector. The source presents this as a conjecture attributed to Kalai; its resolution status is not specified.