Finite-center conjecture for AIP in CAT(0) groups

Let $G$ be a CAT(0) group, meaning a group acting properly and cocompactly by isometries on a CAT(0) space. An Abelian subgroup $A\leq G$ is highest if $A$ does not have a finite-index subgroup contained in an Abelian subgroup of higher rank. The group $G$ has the Abelian intersection property (AIP) if there are highest Abelian subgroups $A_1,\dots,A_n<G$ such that

$$\bigcap_{i=1}^n A_i=\{e\}.$$

Finite-center conjecture. If $G$ is a CAT(0) group and $G$ and every finite-index subgroup of $G$ have finite center, then $G$ has AIP. This would clarify when the AIP hypothesis used in product decomposition results is available; the paper establishes AIP for non-elementary hyperbolic groups, but leaves this CAT(0)-group criterion as a conjecture.