The rough-isometry conjecture for non-special Heintze groups

Let $S$) be a Heintze group that is not among the special-type subgroups of $\operatorname{SO}(n,1)$ or $\operatorname{SU}(n,1)$ for any $n\geqslant 2$. Equip $S$ with any left-invariant Riemannian metric. The rough-isometry conjecture. Every self-quasiisometry of $S$ is a rough isometry. This is supported by the cases established in the paper, but the assertion for all such Heintze groups remains open.