Incompressible-set covering conjecture for higher-rank CAT(0) groups

Let $G$ act geometrically on a CAT(0) space $X$, and let $\mathcal{I}(G)$ denote the family of incompressible subsets introduced in the source. Say that $G$ has higher rank when it does not contain a rank-one isometry.

Incompressible-covering conjecture. If $G$ has higher rank, then

$$\mathcal{I}(G)\text{ covers }\partial X.$$

The conjecture is presented as an approach to the closing lemma: proving this covering property would give the stated higher-rank and Tits-diameter consequences. Its resolution status is not specified.