The uniformly quantitative coarse Baum–Connes conjecture with coefficients

Let $(X_i)_{i\in\mathcal{I}}$ be a family of proper metric spaces. For each $k\leq k'$, let $QI_A(k,k',\varepsilon,r)$ and $QS_A(k,k',\varepsilon,\varepsilon',r,r')$ be the quantitative statements defined using the corresponding Rips complexes and quantitative assembly maps.

Uniformly 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 every $X_i$; 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 every $X_i$.

This uniform version is introduced to reduce the quantitative conjecture for a proper metric space to a sequence of bounded metric spaces. The source does not state a resolution of this uniform conjecture.