The characterization of amenable Hilbert geometries as $G_n$-polygons

Let $(\mathcal{C},d_\mathcal{C})$ be a Hilbert geometry. Recall that it is amenable when, for some separated net $X$ of $\mathcal{C}$, the pseudo-group of bounded perturbations of the identity, $\mathcal{W}(X)$, is amenable. A $G_n$-polygon is a Hilbert geometry of the stated type.

Amenability characterization. A Hilbert geometry is amenable if and only if it is a $G_n$-polygon.

The question seeks a converse to the known fact that every $G_n$-polygon is amenable. The preceding theorem characterizes amenability by the vanishing of the bottom of the spectrum, but the provided text does not establish whether every amenable Hilbert geometry is a $G_n$-polygon.