Let $G$ be a CAT(0) group which acts on the CAT(0) space $X$ properly and cocompactly via isometry. Let $g \in G$ be a hyperbolic isometry of $X$. Then $g$ is called $\textbf{rank one}$ if no axis of $g$ bounds a flat half plane.

$\textbf{Question:}$ Let $G$ be a torsion free CAT(0) group which is not virtually cyclic. Then if $G$ does not have a subgroup isomorphic to $\mathbb{Z}^2$, does $G$ always contain a rank one isometry?

Some remarks:

(1) The question is implied by the flat closing conjecture: $G$ contains $\mathbb{Z}^d$ iff $X$ contains a $d$-flat.

(2) When $X$ is a CAT(0) cube complex, this can be deduced by the rank rigidity theorem for CAT(0) cube groups of Caprace and Sageev in

Rank rigidity for CAT(0) cube complexes. Geom. Funct. Anal. 21 (2011), no. 4, 851–891.

(3) A weaker question one can ask is that does such $G$ always contain a free subgroup?

(4) One can of course also ask the question for any CAT(0) group or more generally groups acting on CAT(0) spaces via semi-simple isometry.