Resolution conjecture for collections of abelian groups

Let ${\cal G}$ be a collection of abelian groups, let $X$ be a compactum, and let $n\geq 2$. Write

$$\overline{\sigma({\cal G})}=\bigcup_{G\in{\cal G}}\overline{\sigma(G)}.$$

A map $r:Z\longrightarrow X$ is $G$-acyclic when it is acyclic with respect to the abelian group $G$.

Resolution conjecture. There exist a compactum $Z$ with $\dim Z\leq n$ and a surjective map $r:Z\longrightarrow X$ such that

$$\dim_G Z\leq\max\{\dim_G X,2\}$$

and $r$ is $G$-acyclic for every $G\in{\cal G}$ if and only if

$$\dim_E X\leq n$$

for every $E\in\overline{\sigma({\cal G})}$.

This conjecture asks for a simultaneous acyclic resolution realizing the cohomological-dimension bounds associated with a collection of coefficient groups. The preceding theorem establishes a version with $\dim Z\leq n+1$ under the corresponding hypotheses; the sharper dimension bound $n$ in the stated equivalence is the unresolved part.