The covariant representation-ring conjecture for infinite groups

Let $G$ be a group and let $F$ be a field of characteristic zero. Define

$$R_{\operatorname{cov},F}(G):={\operatorname{colim}}_{H\in\operatorname{Sub}_{{\mathcal F}\!{\mathcal I}\!{\mathcal N}}(G)}R_F(H),$$

where the colimit uses induction along inclusions of finite subgroups. Covariant representation-ring conjecture. The obvious map

$$W^{G,F}\colon R_{\operatorname{cov},F}(G)\longrightarrow K_0(F[G])$$

is a bijection of abelian groups. This conjecture follows from the Farrell–Jones conjecture for algebraic $K$-theory for $F[G]$; no counterexamples were known at the time of writing.