Strongly collapsible boundary characterization of rank-one CAT(0) groups

Let $G$ act geometrically on a CAT(0) space $X$. A subset of $\partial_\infty X$ is strongly collapsible if it has the dynamical collapse property defined earlier in the source.

Strong-collapse rank-one conjecture. The boundary $\partial_\infty X$ is strongly collapsible if and only if $G$ has rank one.

The source explains that rank one implies strong collapsibility and that affirming the conjecture would make $\mathcal{I}(G)$ the only obstruction to rank one. The converse is left as a conjecture.