The generalized Strong Bass conjecture for rapid decay group algebras

Let $\mathcal{B}$ be a bounding class and let $(G,L)$ be a group with word-length function. Write $\mathcal{H}_{\mathcal{B},L}(G)$ for the associated rapid decay algebra. For a conjugacy class $\langle x\rangle$, call it elliptic if it has finite order and non-elliptic if $x$ has infinite order. The algebra has cyclic homology summands indexed by conjugacy classes, with projection

$$p_{\langle x\rangle}:HC^t_*(\mathcal{H}_{\mathcal{B},L}(G))\twoheadrightarrow HC^t_*(\mathcal{H}_{\mathcal{B},L}(G))_{\langle x\rangle}.$$

Generalized Strong Bass conjecture. For each non-elliptic conjugacy class $\langle x\rangle$, the image of

$$p_{\langle x\rangle}\circ ch_*:K^t_*(\mathcal{H}_{\mathcal{B},L}(G))\to HC^t_*(\mathcal{H}_{\mathcal{B},L}(G))\twoheadrightarrow HC^t_*(\mathcal{H}_{\mathcal{B},L}(G))_{\langle x\rangle}$$

is zero. This generalizes the Strong Bass conjecture by asserting that the Chern character has no contribution in the cyclic-homology summand associated to any non-elliptic conjugacy class. The source gives no resolution, so the conjecture is recorded as open.