Amenable/non-amenable quasi-isometry conjecture for hyperbolic locally compact groups

Let $G$ be a hyperbolic locally compact group, and let $X$ be a metric space. A group is quasi-isometric to another group or space when their large-scale geometries are equivalent. Amenable/non-amenable quasi-isometry conjecture. The group $G$ is quasi-isometric to both an amenable and a non-amenable compactly generated locally compact group if and only if $G$ admits a continuous proper cocompact isometric action on a space $X$ that is either a rank-one symmetric space of non-compact type or a regular tree of valency at least $3$. One implication is established in the source; the converse is presented as conjectural and is equivalent there to more specific conjectures about purely real Heintze groups.