von Stengel's conjecture on estranged facet pairs of simplicial polytopes

Let $P$ be a simplicial $d$-polytope with $2d$ vertices. Two facets are estranged if they are disjoint. Von Stengel's conjecture. The maximum number of pairs of estranged facets of any simplicial $d$-polytope with $2d$ vertices is

$$2^{d-1},$$

which is attained by the $d$-dimensional cross polytope. This conjecture gives a deterministic extremal benchmark for the expected number of estranged facet pairs in random point sets; the paper places its Gaussian estimate in this context, but the supplied text does not state whether the conjecture has been resolved.