The consistent coarse Baum–Connes conjecture with filtered coefficients

Let $A$ be a filtered $C^{\ast}$-algebra, let $X$ be a proper metric space, and let $N_X$ be a locally finite net in $X$. Let $\bigsqcup_{\mathbb{N}}X_i$ denote the separated coarse union of a sequence of metric spaces, and let $P_k(N_X)$ be the $k$-Rips complex. The corresponding Roe and localization algebras with filtered coefficients are denoted by $C_f^{\ast}(\bigsqcup_{\mathbb{N}}P_k(N_X),A)$ and $C_{L,f}^{\ast}(\bigsqcup_{\mathbb{N}}P_k(N_X),A)$. The consistent coarse Baum–Connes conjecture with filtered coefficients. Evaluation at zero induces a homomorphism

$$e_{\ast}: \lim_{k\rightarrow \infty} K_{\ast}(C_{L,f}^{\ast}(\bigsqcup_{\mathbb{N}} P_k(N_X), A)) \longrightarrow \lim_{k\rightarrow \infty} K_{\ast}(C_f^{\ast}(\bigsqcup_{\mathbb{N}} P_k(N_X), A))$$

which is an isomorphism between abelian groups; equivalently, the coarse Baum–Connes conjecture with filtered coefficients in $A$ holds for $\bigsqcup_{\mathbb{N}}X$. The supplied text introduces this conjecture but gives no resolution, so its status remains open.