Triviality of the product map μ2(A) ⊗ H2(E2(A), Z) → H3(E2(A), Z) for local domains

Determine whether, for any local domain A with residue field cardinality |A/mA| ≠ 2, the product map μ2(A) ⊗ H2(E2(A), Z) → H3(E2(A), Z) induced by the action of μ2(A) on E2(A) via (-1, X) ↦ -X is trivial.

Background

This question arises in the context of comparing third homology groups of E2(A) and PE2(A) with refined Bloch groups. The product map from μ2(A) ⊗ H2(E2(A), Z) to H3(E2(A), Z) is central in constructing refined Bloch–Wigner exact sequences. The authors note that answering this triviality question positively would help establish a projective Bloch–Wigner exact sequence and, in turn, contribute to resolving Question 6.4.

The paper proves triviality of certain components of this product under additional hypotheses (e.g., −1 ∈ (A×)2 or |GA| ≤ 4), but explicitly states that the general case for local domains with |A/mA| ≠ 2 remains unresolved, even over infinite fields.