Universal acyclic resolutions for finitely generated coefficient groups

We prove that for every compactum X and every integer n≥2 there are a compactum Z of dim≤n and a surjective UV^n-1-map r :Z→X having the property that: for every finitely generated Abelian group G and every integer k≥2 such that dim_GX≤k≤n we have dim_GZ≤k and r is G-acyclic, or equivalently: for every simply connected CW-complex K with finitely generated homotopy groups such that e-dimX≤K we have e-dimZ≤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.)