Classification conjecture for polycyclic abelian-by-cyclic groups

Let $\Gamma_M$ and $\Gamma_N$ be polycyclic abelian-by-cyclic groups associated to matrices $M$ and $N$ satisfying

$$\left|\det M\right|,\left|\det N\right|=1.$$

For a real exponent, write $M^a=\phi(a)$, where $\phi:\mathbb R\to\operatorname{GL}(n,\mathbb R)$ is a one-parameter subgroup with $\phi(1)=M$; likewise for $N^b$.

Classification conjecture. The groups $\Gamma_M$ and $\Gamma_N$ are quasi-isometric if and only if there exist nonzero $a,b\in\mathbb R$ such that $M^a$ and $N^b$ have the same absolute Jordan form.

This predicts that quasi-isometry classes in the polycyclic case are determined by absolute Jordan form up to real powers, contrasting with the finer integral-power behavior in the nonpolycyclic case. The supplied status marks this conjecture as open.