Burghelea's conjecture for periodic cyclic homology

Let $G$ be a group. Write $\langle G \rangle^{\infty}$ for the conjugacy classes of infinite-order elements, and for $g\in G$ let $C_g$ be its centralizer and $T^G_\ast(g;\mathbb{C})$ the inverse-limit term in Burghelea's decomposition of periodic cyclic homology. Burghelea's conjecture. The group

$$\bigoplus_{[g] \in \langle G \rangle^{\infty}} T^G_\ast(g;\mathbb{C})$$

vanishes. Burghelea constructed a counterexample, so the conjecture is false for arbitrary groups; it is known for several classes, including hyperbolic and arithmetic groups.