Burghelea's conjecture on periodic cyclic homology of group algebras

Let $G$ be a discrete group, let $K(G,1)$ be an Eilenberg–Mac Lane space, and let $\langle G\rangle^{\infty}$ denote the conjugacy classes of elements of infinite order. For $x\in\langle G\rangle^{\infty}$, write $G_x$ for the centralizer of $x$, $N_x=G_x/\langle x\rangle$ for the reduced centralizer, and let $T_*(x,\mathbb{Q})$ be the term in Burghelea's decomposition of the periodic cyclic homology of $\mathbb{Q}G$, fitting into the short exact sequence

$$0\to\varprojlim{}^1\{H_{*-1+2n}(N_x,\mathbb{Q})\}\to T_*(x,\mathbb{Q})\to\varprojlim\{H_{*+2n}(N_x,\mathbb{Q})\}\to 0.$$

Burghelea's conjecture. If $K(G,1)$ has the homotopy type of a finite CW-complex, then

$$T_*(x,\mathbb{Q})=0$$

for all $x\in\langle G\rangle^{\infty}$. Burghelea's formula expresses periodic cyclic homology in terms of these groups for infinite-order conjugacy classes; the conjecture asserts their vanishing under a finiteness hypothesis on $K(G,1)$, and the supplied source gives no resolution.