Path-disconnected boundaries for iterated amalgams of CAT(0) groups

Let $G_-$ and $G_+$ be infinite CAT(0) groups, and let $m,n$ be positive integers. Define

$$G=(G_-\times \mathbb{Z}^n)\ast_{\mathbb{Z}^n}(\mathbb{Z}^n\times\mathbb{Z}^m)\ast_{\mathbb{Z}^m}(\mathbb{Z}^m\times G_+).$$

A CAT(0) space on which $G$ acts is denoted by $X$, with visual boundary $\partial X$.

Generalized boundary conjecture. The group $G$ acts on a CAT(0) space $X$ such that $\partial X$ is not path connected.

The paper proves this assertion in two special cases, with $(n,m)=(1,1)$ and $(n,m)=(1,2)$. The conjecture extends that result to the displayed family of iterated amalgams.