Higher Rank Flat Closing conjecture

Let $X$ be a locally compact geodesically complete CAT(0) space with Tits diameter $\pi$, and suppose that $X$ supports a geometric group action $\Gamma\curvearrowright X$. If $X$ has rank $n$, meaning that $n$ is the maximal dimension of an isometrically embedded Euclidean flat, then $X$ contains a $\Gamma$-periodic $n$-flat. Higher Rank Flat Closing conjecture. Under these hypotheses, $X$ contains a periodic $n$-flat. This is a special case of Gromov's Flat Closing Problem. The paper notes that it holds for rank $2$ spaces of Tits diameter $\pi$ and for Riemannian symmetric spaces and Euclidean buildings, but the stated higher-rank case remains open.