Integral-homology version of the stable splitting for Hom(Z^n, U(m))

Ascertain whether the stable splitting of Hom(Z^n, U(m)) established after inverting m! induces an isomorphism on integral homology; specifically, prove or refute that H_*(Hom(Z^n, U(m)); Z)_+ ≅ H̃_*(U(m) ∧_+ ⨁_{(λ_a)∈P} (∧_{a∈I} U(λ_a) C_{|a|}(u_{λ_a})^+); Z), where I = {0,1}^n, P is the set of I-indexed partitions of m, and C_{|a|}(u_{λ_a}) denotes the commuting variety in the Lie algebra u_{λ_a}.

Background

The main theorem provides a stable splitting of Hom(Z^n, U(m)) after inverting m!, yielding a decomposition into wedge summands built from commuting varieties over partial flag manifolds. However, this localization removes torsion information.

The authors ask whether an analogous statement holds at the level of integral homology. They formulate this explicitly as Question C and note preliminary evidence in small cases (n = 2, m = 2,3), making this a concrete open problem about integral homology isomorphism.