The horizontal-preserving conjecture for polycyclic abelian-by-cyclic groups

Let $M$) and $N$ be matrices with

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

Assume that $M$ and $N$ have no eigenvalues on the unit circle, and let $G_M$ and $G_N$ be the associated abelian-by-cyclic Lie groups. A quasi-isometry $G_M\to G_N$ is called horizontal-respecting when it preserves the horizontal foliations.

Horizontal-preserving conjecture. Every quasi-isometry $G_M\to G_N$ is horizontal-respecting.

This is the key rigidity assertion in the polycyclic abelian-by-cyclic case and is intended to lead to the classification by absolute Jordan form. The source identifies it as an open problem; the supplied context does not state that it has been resolved.