Rigidity conjecture for the bottom of the spectrum of regular Hilbert geometries

Let $(\mathcal{C},d_{\mathcal{C}})$ be a regular Hilbert geometry, and let $\lambda_1(\mathcal{C})$ denote the bottom of the spectrum of its Finsler Laplacian. Rigidity conjecture. The equality

$$\lambda_1(\mathcal{C}) = \frac{(n-1)^2}{4}$$

holds if and only if $\mathcal{C}$ is an ellipsoid. This asks whether equality in the spectral lower bound characterizes hyperbolic geometry among regular Hilbert geometries; the surrounding discussion notes that the equality occurs for hyperbolic space and asks whether every non-ellipsoidal regular Hilbert geometry has an eigenvalue below $\frac{(n-1)^2}{4}$.