The volume entropy bound for Hilbert geometries

Let $\Omega\subset\mathbb{R}\mathbb{P}^n$ be a convex proper open set, and let $h_{vol}(\Omega,d_{\Omega})$ denote the volume entropy of its Hilbert geometry, defined using the Busemann volume. Volume entropy conjecture. For any $\Omega\subset\mathbb{R}\mathbb{P}^n$,

$$h_{vol}(\Omega,d_{\Omega})\leqslant n-1.$$

This conjecture asserts that the volume entropy of every Hilbert geometry is bounded above by the entropy of the ellipsoid, while polytopes provide the opposite extremal case with zero volume entropy. It is proved in dimension $n=2$, and examples are known with $0<h_{vol}<1$; the general conjecture remains open.