The quasi-isometric restriction conjecture for linear semidirect products

Let $\Gamma$ be a finitely generated group, and let $\rho$ be a linear representation of $\Gamma$ on $\mathbf{R}^n$. Equip $\mathbf{R}^n\rtimes_\rho\Gamma$ and $\mathbf{R}^n\rtimes\overline{\rho(\Gamma)}$ with word lengths associated to compact generating subsets, and restrict these word lengths to the respective copies of $\mathbf{R}^n$. Quasi-isometric restriction conjecture. The canonical homomorphism

$$\mathbf{R}^n\rtimes_\rho\Gamma \longrightarrow \mathbf{R}^n\rtimes\overline{\rho(\Gamma)}$$

is a quasi-isometry when restricted to $\mathbf{R}^n$. This question arises in the study of the metric geometry of semidirect products associated with generalized Baumslag–Solitar groups; only partial results are available in the source.