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Baum-Connes conjecture states that for a locally compact group $G$ the so called assebly map $\mu$ between $G$-equivariant K-homology of the universal example for proper actions of $G$ and K-theory of $C^*_r(G)$ (the reduced $C^*$-algebra of the group) is an isomorphism. Is there an example of the group $G$ such that:
1. It is not known whether Baum-Connes conjecture for this group holds but
2. Groups $K_G(\underline{E}G)$ and $K(C^*_r(G))$ are abstracly isomorphic and nontrivial.

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    $\begingroup$ I don't know, but I'd be a little surprised - there aren't actually that many techniques for calculating $K(C_r^*(G))$ other than the Baum-Connes conjecture. $\endgroup$
    – Paul Siegel
    Commented Oct 10, 2017 at 16:20

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