Generalized Burghelea Conjecture

Let $G$ be a discrete group. For an element $x$ in $G$, let $G_x$ denote its centralizer, let $N_x=G_x/\langle x\rangle$ be the reduced centralizer, and let ${\langle G\rangle}^{\infty}$ denote the set of conjugacy classes of elements of infinite order. The group $T_*(x,\mathbb Q)$ is the term associated to an infinite-order conjugacy class in the periodic cyclic homology decomposition of $\mathbb QG$. Generalized Burghelea Conjecture. For every $x\in{\langle G\rangle}^{\infty}$, one has

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

The conjecture asserts the vanishing of the infinite-order contributions in Burghelea's computation of the periodic cyclic homology of a group algebra. The source paper constructs a finitely generated group that does not satisfy the Burghelea conjecture, so this statement is refuted in the generality given here.