Hamenstädt's quasi-isometric rigidity conjecture for purely real Heintze groups

A purely real Heintze group is a simply connected solvable Lie group of the form $N\rtimes\mathbf{R}$, where $N$ is simply connected and nilpotent and the action of $\mathbf{R}$ has only real eigenvalues. Hamenstädt's conjecture. Let $H$ be a purely real Heintze group. If a purely real Heintze group $L$ is quasi-isometric to $H$, then $L$ is isomorphic to $H$; equivalently, two purely real Heintze groups are quasi-isometric if and only if they are isomorphic. The conjecture is known for abelian groups, groups associated with rank-one symmetric spaces, and groups of Carnot type, but remains open in general.