A-theoretic isomorphism conjecture for families of groups

Let $\cF$ be a family of groups, let $G$ be a countable discrete group, and let $W$ be a free $G$-CW-complex. Let $E_\cF G$ denote the classifying space of $G$ for the family $\cF$, and let $\Aa_W^{-\infty}$ denote the associated non-connective $A$-theory coefficient spectrum. A-theoretic isomorphism conjecture. The assembly map

$$\alpha_{\cF,W}\colon \HH^G(E_\cF G;\Aa_W^{-\infty})\to \HH^G(G/G;\Aa_W^{-\infty})\cong \Aa^{-\infty}(G\backslash W)$$

is a weak equivalence.

This is the isomorphism conjecture for $A$-theory, formulated for arbitrary families of groups and free equivariant CW-complexes. The source records it as a conjectural statement; no resolution is supplied in the given text.