The conjecture on a non-rigid CAT(0) group with uncountably many boundaries

Let $F_2$ be the free group of rank $2$, let $\mathbb{Z}$ and $\mathbb{Z}_2$ denote the infinite cyclic group and the cyclic group of order $2$, respectively, and set

$$G=(F_2\times\mathbb{Z})*\mathbb{Z}_2.$$

Non-rigidity conjecture. The group $G$ will be a non-rigid CAT(0) group with uncountably many boundaries.

This conjecture arises from examples and applications concerning rigidity of boundaries of CAT(0) groups. It predicts both failure of rigidity and the existence of uncountably many distinct boundaries for this particular group; no resolution is given in the supplied text.