The Strong Bass conjecture

Let $G$ be a group, let $r \colon K_0\mathbb{Z}G\rightarrow HH_0(\mathbb{Z}G)$ be the Hattori–Stallings trace map, and write the image of a finitely generated projective module $P$ as

$$r(P)=\sum_{[g]\in \operatorname{Conj}(G)}r_P(g).$$

Here $r_P(g)$ is the coefficient associated to the conjugacy class of $g$. Strong Bass conjecture. The function $r_P(g)$ is $0$ for $g\neq 1$. This conjecture concerns the vanishing of nonidentity contributions to the Hattori–Stallings trace. It follows from the stronger claim that the reduced map from $\widetilde{K_0\mathbb{Z}G}$ to $\widetilde{K_0\mathbb{Q}G}$ vanishes rationally; the latter is known when $G$ satisfies the Farrell–Jones conjecture, but the general case remains open.