Maximal-matching Hamilton-path characterization

Prove that, for n ≥ 5 and a maximal matching M of Q_n with uv not in M, a Hamilton path with endpoints u and v extending M exists if and only if the matching (M \setminus {uu^M, vv^M}) \cup {u^M v^M} contains no half-layer.

Background

This is a proposed simplification of the general Hamilton-path conjecture for maximal matchings. It parallels the known characterization for perfect matchings in the complete bipartite graph associated with the hypercube. The paper proves that the general C-condition conjecture would imply this maximal-matching version, but does not establish the conjecture independently in all dimensions.