Quasi-isometry group conjecture for Solv geometry

Let $\operatorname{Solv}$ be the three-dimensional Solv Lie group, whose two boundary factors are each identified with $\mathbb R$. Let $\operatorname{Bilip}(\mathbb R)$ denote the group of bilipschitz homeomorphisms of $\mathbb R$, and let $\mathbb Z/2$ act by switching the two factors.

QI-group conjecture.

$$\operatorname{QI}(\operatorname{Solv})=\bigl(\operatorname{Bilip}(\mathbb R)\times\operatorname{Bilip}(\mathbb R)\bigr)\rtimes\mathbb Z/2.$$

The conjecture is motivated by the boundary description of quasi-isometries, analogous to the characterization of quasi-isometries of hyperbolic space by boundary maps. The source gives evidence for one inclusion and records implications relating this conjecture to the horizontal-preserving and rigidity conjectures; the supplied status marks it as resolved.