The Meta-Isomorphism Conjecture for functors from spaces to spectra

Let ${\mathbf S} \colon \operatorname{SPACES}\to \operatorname{SPECTRA}$ be a covariant functor respecting weak equivalences and disjoint unions. For a group $G$, let ${\mathcal F}$ be a family of subgroups and let $E_{\mathcal F}(G)$ be the classifying $G$-CW-complex for this family. The associated equivariant homology theory is denoted $H_*^G(-;{\mathbf S}^B)$, with $H_n^G(G/G;{\mathbf S}^B)\cong\pi_n({\mathbf S}(BG))$. Meta-Isomorphism Conjecture for ${\mathbf S}$. The assembly map induced by the projection $\operatorname{pr}\colon E_{\mathcal F}(G)\to G/G$ is bijective for all $n\in{\mathbb Z}$:

$$H_n^G(\operatorname{pr};{\mathbf S}^B)\colon H_n^G(E_{\mathcal F}(G);{\mathbf S}^B)\to H_n^G(G/G;{\mathbf S}^B)\cong\pi_n({\mathbf S}(BG)).$$

This is the general assembly-map formulation underlying the later $A$-theory Farrell--Jones conjectures; the supplied text gives no resolution status.