Hamilton-path extension from an uncovered vertex

Establish that, for every matching M of the hypercube Q_n and every pair of opposite-parity vertices u and v with u uncovered by M, there exists a Hamilton path extending M and having u and v as its endpoints.

Background

This conjecture is a strengthening of the unrestricted matching-to-Hamilton-cycle problem. If one endpoint is uncovered, a Hamilton path extending the matching would immediately yield a Hamilton cycle by adding an edge incident with the uncovered endpoint. The paper verifies the conjecture computationally in dimensions 2 through 5 and uses it conditionally to derive cycle-extension results for matchings spanning at most d directions.