Planar boundary conjecture for CAT(0) groups

Let $G$ be a $\operatorname{CAT}(0)$ group with a planar visual boundary. A visual boundary of $G$ is a boundary at infinity associated with a proper $\operatorname{CAT}(0)$ space on which $G$ acts geometrically.

Planar boundary conjecture. Every visual boundary of $G$ is planar, and furthermore, $G$ is virtually the fundamental group of a compact 3-manifold.

This conjecture proposes a converse to known implications from geometric 3-manifold groups and Kleinian groups to planarity of CAT(0) boundaries. The paper develops evidence for it in the setting of right-angled Coxeter groups; its general status is not established here.