The algebraic–geometric rank equivalence for CAT(0) groups

Let $X$ be a complete $\mathrm{CAT}(0)$ space and let $G$ be a group acting geometrically on $X$, meaning properly and cocompactly by isometries. A geodesic line in $X$ has rank one if it does not bound a flat half-plane, and $X$ has higher geometric rank if no geodesic in $X$ has rank one. The group $G$ has higher algebraic rank in the sense of Prasad and Raghunathan. Algebraic–geometric rank conjecture. $G$ has higher algebraic rank if and only if $X$ has higher geometric rank. This asks for a CAT(0)-space analogue of the relationship between algebraic and geometric rank established for complete Riemannian manifolds of bounded nonpositive sectional curvature and finite volume. The supplied text does not indicate whether the equivalence has been proved or disproved.