The maximal entropy conjecture for Hilbert geometries

Let $K$ be an $n$-dimensional convex set equipped with its Hilbert geometry. Fix a volume and, for a metric ball $B(o,r)$, define the upper volume entropy by

$$\overline{\operatorname{Ent}} K:=\limsup_{r\to\infty}\frac{\log\operatorname{Vol} B(o,r)}{r}.$$

The hyperbolic space has entropy $n-1$. Maximal entropy conjecture. For any $n$-dimensional Hilbert geometry,

$$\overline{\operatorname{Ent}} K\leq n-1.$$

This conjecture asserts that hyperbolic space, equivalently an ellipsoid in Hilbert geometry, has maximal entropy among Hilbert geometries of the same dimension. The ordinary entropy limit need not be known to exist in general, so the conjecture is formulated using the upper entropy; it remains open in the stated generality.