The localization-algebra formulation of the Baum–Connes conjecture with coefficients

Let $\Gamma$ be a discrete countable group with a length function $|\cdot|$, let $P_d(\Gamma)$ be its Rips complex at scale $d$, and let $C^{\ast}(P_d(\Gamma),A)^{\Gamma}$ and $C_L^{\ast}(P_d(\Gamma),A)^{\Gamma}$ denote the equivariant Roe and localization algebras with coefficients in the $\Gamma$-$C^{\ast}$-algebra $A$. The Baum–Connes conjecture with coefficients in $A$ for $\Gamma$ states that the evaluation-at-zero homomorphism

$$e_{\ast}: \lim_{d\rightarrow \infty} K_{\ast}(C^{\ast}_{L}(P_{d}(\Gamma), A)^{\Gamma}) \rightarrow \lim_{d\rightarrow \infty} K_{\ast}(C^{\ast}(P_{d}(\Gamma), A)^{\Gamma})$$

is an isomorphism. This is a localization-algebra model for the Baum–Connes assembly map, expressing the conjecture through the direct-limit $K$-theory of Rips complexes. The supplied text does not state whether this formulation is open or resolved in general.