The torsion conjecture for discrete CAT(0)-space isometry groups

Let $X$ be a proper, $\operatorname{CAT}(0)$-space and let $\Gamma$ be a discrete group of isometries of $X$, not necessarily cocompact. A group is torsion if it contains only finite-order elements.

Torsion conjecture. If $\Gamma$ is torsion, then $\Gamma$ is finite. An easier version is that if $\Gamma$ is almost-cocompact, then it contains an infinite-order element.

The conjecture concerns whether properness and discreteness impose finiteness on torsion isometry groups; the paper presents the almost-cocompact assertion as the version needed for its application. No resolution is given in the supplied context.