Horizontal-respecting conjecture for quasi-isometries of Solv

Let $\operatorname{Solv}$ denote the three-dimensional Solv Lie group. A quasi-isometry $f:\operatorname{Solv}\to\operatorname{Solv}$ is called horizontal-respecting when it preserves the horizontal foliation.

Horizontal-respecting conjecture. Every quasi-isometry

$$f:\operatorname{Solv}\to\operatorname{Solv}$$

is horizontal-respecting.

This is the two-dimensional-matrix specialization of the horizontal-preserving conjecture and is linked in the source to the conjectured description of the quasi-isometry group of $\operatorname{Solv}$. The supplied status is unknown, so the database records it as open.