The standard-form conjecture for quasi-isometries of nondegenerate Sol-type groups

Let $G\cong \prod_{i=1}^n \mathbf{N}_i \rtimes \mathbb{R}^k$ be nondegenerate with $n\ge 2$. Let $q:G\to G$ be a quasi-isometry for some, and hence any, left-invariant Riemannian metric. Let $\overline{\mathbb{R}^k}$ be any lift of the quotient map $G\to\mathbb{R}^k$, and give $G$ semi-direct normal coordinates with the factor $\overline{\mathbb{R}^k}$ on the right. Standard-form conjecture. The map $q$ is at finite distance from a map $L_x\circ (\prod_{i=1}^n f_i)\circ\sigma$, where $L_x$ is left-translation by a group element, each $f_i$ is a map from $\mathbf{N}_i$ to itself, and $\sigma$ belongs to a finite group of symmetries that simultaneously permute the $\mathbf{N}_i$ and act on $\overline{\mathbb{R}^k}$ by a linear isomorphism permuting the associated roots, with the property that if $\sigma^*\alpha_i=\alpha_j$, then $\sigma(\mathbf{N}_i)=\mathbf{N}_j$ and $(\mathbf{N}_i,D_i)$ is isomorphic to $(\mathbf{N}_j,D_j)$. The conjecture would give the structural description of quasi-isometries needed to establish intrinsic rough-isometry results for these groups; the supplied text does not indicate whether it is known or open.