The quasi-isometric classification conjecture for polycyclic abelian-by-cyclic groups

Let $M\in\operatorname{SL}(n,\mathbf{Z})$ have no eigenvalues on the unit circle, and let $G$ be a finitely generated group quasi-isometric to the associated group $\Gamma_M$. The quasi-isometric classification conjecture. There is a finite normal subgroup $F\triangleleft G$ such that $G/F$ is abstractly commensurable to $\Gamma_N$ for some $N\in\operatorname{SL}(n,\mathbf{Z})$ with no eigenvalues on the unit circle. Together with the preceding theorem and conjecture, this would give a quasi-isometric classification of these polycyclic groups.