Rank Rigidity Conjecture for proper geodesically complete CAT(0) spaces

Let $X$ be a proper, geodesically complete $\operatorname{CAT}(0)$ space under a geometric action. A rank one axis is an axis in $X$ that does not bound a flat half-plane. Rank Rigidity Conjecture. If $X$ does not admit a rank one axis, then $X$ is a higher rank symmetric space or Euclidean building, or splits as a nontrivial product.

This conjecture generalizes the rank rigidity theorem for compact nonpositively curved manifolds to the metric setting of $\operatorname{CAT}(0)$ spaces. It is known for several classes with additional Riemannian or polyhedral structure, but remains open for general proper geodesically complete $\operatorname{CAT}(0)$ spaces.