Finite-part assembly-map conjecture

Let $\Gamma$ be a countable discrete group. For each nontrivial finite-order element $\gamma\in\Gamma$, let $p_\gamma$ denote the associated $K$-theory class, and let $K_0^{\mathrm{fin}}(C^*(\Gamma))$ be the subgroup generated by these classes. Let $\widetilde K_0^{\mathrm{fin}}(C^*(\Gamma))$ be its quotient by the subgroup generated by differences $[p_\gamma]-[p_\beta]$ whenever $\gamma$ and $\beta$ have the same order. For nontrivial elements $\gamma_1,\ldots,\gamma_n$ of distinct finite orders, let $\widetilde{\mathcal M}_{\gamma_1,\ldots,\gamma_n}$ be the image of the subgroup generated by $[p_{\gamma_1}],\ldots,[p_{\gamma_n}]$ in this quotient. Let $E\Gamma$ be the universal space for a free proper $\Gamma$-action, and let

$$\mu:K_0^\Gamma(E\Gamma)\longrightarrow K_0(C^*(\Gamma))$$

be the assembly map.

Finite-part assembly conjecture. The group $\widetilde{\mathcal M}_{\gamma_1,\ldots,\gamma_n}$ has rank $n$, and every nonzero element of $K_0^{\mathrm{fin}}(C^*(\Gamma))$ lies outside the image of $\mu$.

The preceding proposition establishes both assertions when $\Gamma$ is finitely embeddable into Hilbert space. The conjecture asks for these properties for arbitrary countable discrete groups.