The ℓ¹ Bass conjecture

Let $G$ be a discrete group, let $[G]$ be its set of conjugacy classes, and let $\mathrm{FC}(G)$ be the subset consisting of conjugacy classes represented by elements of finite order. The $\ell^1$ Hattori–Stallings trace is

$$HS^1:K_0(\ell^1(G))\longrightarrow\ell^1([G]).$$

The $\ell^1$ Bass conjecture. The image of $HS^1$ is contained in

$$\bigoplus_{\mathrm{FC}(G)}C{C},$$

the subspace of functions finitely supported on conjugacy classes of elements of finite order. The paper proves that the Bost conjecture implies this statement, while its general validity is not resolved in the supplied text.