Cornulier's pointed-sphere conjecture for quasiisometries of non-special Heintze groups

Let $S$ be a Heintze group, and let $\phi$ be a self-quasiisometry of $S$. The Gromov boundary of $S$ is the boundary on which $\phi$ induces an extension. Assume that $S$ is not of special type, and let $\xi$ be the unique boundary point fixed by all left-translations of $S$. Cornulier's pointed-sphere conjecture. The extension of $\phi$ to the Gromov boundary of $S$ fixes $\xi$. This boundary-point rigidity is identified as a common ingredient in quasiisometric rigidity and the paper's rough-similarity theorem, but remains conjectural in the stated generality.