The Isomorphism and Fibered Isomorphism conjectures for equivariant homology theories

Let ${\mathcal H}^?_*$ be an equivariant homology theory with values in $R$-modules, let $G$ be a group, and let ${\mathcal C}$ be a family of subgroups of $G$. Let $E_{\mathcal C}(G)$ be the classifying $G$-CW complex for the family ${\mathcal C}$, and let $p:E_{\mathcal C}(G)\to pt$ be the projection. Isomorphism and Fibered Isomorphism conjectures. The Isomorphism conjecture for $(G,\mathcal C)$ states that

$${\mathcal H}^G_n(p):{\mathcal H}^G_n(E_{\mathcal C}(G))\simeq {\mathcal H}^G_n(pt)$$

for every $n\in{\mathbb Z}$. The Fibered Isomorphism conjecture for $(G,\mathcal C)$ states that, for every group homomorphism $\phi:K\to G$, the Isomorphism conjecture holds for $(K,\phi^*{\mathcal C})$.

These conjectures formulate the assembly-map isomorphism expected in equivariant homology theories and include the fibered version, which requires compatibility with pullbacks of families along arbitrary group homomorphisms. The supplied text gives the definitions but no evidence resolving the conjectures in this generality.