Perfect matchings in antipodally colored lattice of subsets

Let P(S) be the family of all subsets of a finite set S. A 2-coloring of P(S) is antipodal if every subset is colored differently than its complement. Is it true that there is a perfect matching between the color classes such that every matched pair is inclusion related? We give a positive answer if the color classes are assumed to be monotone. This answers a question posed by Mazur in connection to a number theoretic problem.