Nonexistence conjecture for upper-half-space quasi-isometric embeddings in Solv

Let $\mathbb H^2$ be the hyperbolic plane and let $\operatorname{Solv}$ be the three-dimensional Solv geometry with coordinates $(x,y,t)$. Define its upper half-space by

$$\operatorname{Solv}^+=\{(x,y,t)\in\operatorname{Solv}:t\geq 0\}.$$

Upper-half-space embedding conjecture. There does not exist a quasi-isometric embedding

$$\mathbb H^2\to\operatorname{Solv}$$

whose image is entirely contained in $\operatorname{Solv}^+$.

This conjecture would support the proposed understanding of quasi-isometrically embedded hyperbolic planes in $\operatorname{Solv}$ and follows the source's discussion of quasivertical embeddings. Its resolution status is not supplied and is therefore recorded as open.