Universal acyclic resolutions for finitely generated coefficient groups

We prove that for every compactum X and every integer $n\ge 2$ there are a compactum Z of $\dim \le n$ and a surjective $U{V}^{n-1}$-map $r:Z\text{lo}X$ having the property that: for every finitely generated abelian group G and every integer $k\ge 2$ such that ${\dim }_{G}X\le k\le n$ we have $\dim_G Z \leq k$ and r is G-acyclic, or equivalently: for every simply connected CW-complex K with finitely generated homotopy groups such that $\text{edim}X\le K$ we have $\text{edim}Z\le K$ and r is K-acyclic. (A space is K-acyclic if every map from the space to K is null-homotopic. A map is K-acyclic if every fiber is K-acyclic.)