Weinberger–Yu's finite K-theory conjecture for group C*-algebras

Let $\Gamma$ be a countable discrete group. For each nontrivial finite-order element $\gamma\in\Gamma$ of order $d$, let

$$p_\gamma=\frac{1}{d}\sum_{k=1}^{d}\gamma^k.$$

Let $K_0^{\mathrm{fin}}(C^*(\Gamma))$ be the subgroup of $K_0(C^*(\Gamma))$ generated by the classes $[p_\gamma]$. For nontrivial elements $\gamma_1,\ldots,\gamma_n$ of distinct finite orders, let $\mathcal M_{\gamma_1,\ldots,\gamma_n}$ be the subgroup generated by $[p_{\gamma_1}],\ldots,[p_{\gamma_n}]$. Let $E\Gamma$ be the universal space for proper and free $\Gamma$-actions.

Weinberger–Yu's conjecture. The group $\mathcal M_{\gamma_1,\ldots,\gamma_n}$ has rank $n$, and every nonzero element of it lies outside the image of the assembly map

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

The conjecture predicts independent finite-order classes in maximal group C*-algebra K-theory and their detection outside the assembly-map image. Its status is not resolved in the supplied source.