The quantitative coarse Baum–Connes conjecture with coefficients

Let $X$ be a proper metric space, let $N_X$ be a locally finite net in $X$, and write $P_k=P_k(N_X)$ for its Rips complex. For $k\leq k'$, let $QI_A(k,k',\varepsilon,r)$ and $QS_A(k,k',\varepsilon,\varepsilon',r,r')$ be the quantitative statements defined using the localization and quantitative Roe algebras, with $0<\varepsilon\leq\varepsilon'<1/4$ and $r+1\leq r'$.

Quantitative coarse Baum–Connes conjecture with coefficients. For every $k\in\mathbb{N}$, there exists $0<\varepsilon<1/4$ such that, for every $r>0$, there exists $k'\geq k$ for which $QI_A(k,k',\varepsilon,r)$ holds for $X$; and there exists $k'\geq k$, $r'\geq r+1$, and $1/4>\varepsilon'\geq\varepsilon$ for which $QS_A(k,k',\varepsilon,\varepsilon',r,r')$ holds for $X$.

This quantitative refinement of the coarse Baum–Connes conjecture is the main notion introduced in the paper. The source proves that the ordinary coarse Baum–Connes conjecture with coefficients implies it, yielding many examples, but does not claim that the conjecture is resolved in general.