Ballmann–Buyalo closing lemma for CAT(0) groups

Let $G$) act properly discontinuously by isometries on a complete CAT(0) space $X$, also called a Hadamard space. Write $\Lambda G$ for the limit set of $G$ and let $\partial_{\mathrm{T}}X$ denote the Tits boundary.

Closing lemma. If $\Lambda G=\partial X$ and

$$\operatorname{diam}(\partial_{\mathrm{T}}X)>\pi,$$

then $X$ contains a $G$-periodic rank-one geodesic.

This is one of the two rank-rigidity problems attributed to Ballmann and Buyalo. Its resolution status is not specified in the source.