Meta-isomorphism conjecture for assembly maps

Fix a group $G$, a family of subgroups $\mathcal{F}$, and a $G$-homology theory $\mathcal{H}^G_*$. Let $E_{\mathcal{F}}(G)$ be the classifying $G$-space for the family $\mathcal{F}$, and let

$$A_{\mathcal{F}}\colon\mathcal{H}^G_n(E_{\mathcal{F}}(G))\to\mathcal{H}^G_n(\operatorname{pt})$$

be the assembly map induced by the projection $E_{\mathcal{F}}(G)\to\operatorname{pt}$.

Meta-isomorphism conjecture. The assembly map $A_{\mathcal{F}}$ is an isomorphism for every $n\in\mathbb{Z}$.

The source notes that the assertion is automatic for the family of all subgroups; the content is to choose the smallest possible family for which it remains true. Farrell–Jones and Baum–Connes conjectures are presented as instances.