Quasi-isometric rigidity conjecture for purely real Heintze groups

A purely real Heintze group is a Heintze group $N\rtimes_\alpha\mathbb{R}$ whose defining derivation $\alpha$ has only real eigenvalues. Quasi-isometric rigidity conjecture. Two purely real Heintze groups are quasi-isometric if, and only if, they are isomorphic. This conjecture predicts that the quasi-isometry class completely determines a purely real Heintze group up to isomorphism. It remains open in full generality, although partial results are known.