Perfect matchings extending on subcubes to Hamiltonian cycles of hypercubes

Recently, Fink [J. Fink, Perfect matchings extend to Hamilton cycles in hypercubes, J. Combin. Theory Ser. B 97 (2007) 1074-1076] affirmatively answered Kreweras' conjecture asserting that every perfect matching of the hypercube extends to a Hamiltonian cycle. We strengthen this result in the following way. Given a partition of the hypercube into subcubes of nonzero dimensions, we show for every perfect matching of the hypercube that it extends on these subcubes to a Hamiltonian cycle if and only if it interconnects them.