Quasi-isometric invariance of irreducibility and atoroidality

Let $F$ and $F'$ be free groups, and let

$$\phi:F\to F,\qquad \psi:F'\to F'$$

be automorphisms. Their mapping tori are the semidirect products $F\rtimes_\phi\mathbb Z$ and $F'\rtimes_\psi\mathbb Z$. Two groups are quasi-isometric if there is a quasi-isometry between their associated metric spaces.

Quasi-isometric invariance of irreducibility and atoroidality. If $F\rtimes_\phi\mathbb Z$ and $F'\rtimes_\psi\mathbb Z$ are quasi-isometric, then $\phi$ is irreducible and atoroidal if and only if $\psi$ is irreducible and atoroidal.

This asserts that the combined dynamical property of being irreducible and atoroidal is a geometric invariant of the mapping torus. The source presents this as a conjectural extension of the proved quasi-isometric invariance of being induced by a pseudo-Anosov; its resolution is not specified here.