Fibered homotopy K-theory isomorphism conjecture

Let $G$ be a group, $R$ a ring, and $H^?_*(-;\mathbf K\mathbf H_R)$ the equivariant homology theory whose value on a group $G$ has homotopy groups $\mathit{KH}_n(RG)$. Let $\mathcal{FIN}$ be the family of finite subgroups of $G$. The pair $(G,\mathcal{FIN})$ is considered with the corresponding fibered and unfibered isomorphism-conjecture formulations.

Fibered homotopy K-theory isomorphism conjecture. The group $G$ satisfies the (fibered) $\mathit{KH}$-isomorphism conjecture for $R$ if the pair $(G,\mathcal{FIN})$ satisfies the (fibered) isomorphism conjecture for the equivariant homology theory $H^?_*(-;\mathbf K\mathbf H_R)$.

This is the homotopy-$K$-theory analogue of the Farrell–Jones isomorphism conjecture. The paper proves that groups acting on trees inherit this conjecture from their isotropy groups, but the conjecture itself is not asserted to be resolved in general.