Matchings in the hypercube with specified edges

Given a matching $M$ in the hypercube $Q^n$, the \emph{profile} of $M$ is the vector $\mathit{x}=(x_1,\dots ,x_n)\in {\mathbb{N}}^n$ such that $M$ contains $x_i$ edges whose endpoints differ in the $i$th coordinate. If $M$ is a perfect matching, then it is clear that $\mathrm{\mid }\mathrm{\mid }\mathit{x}\mathrm{\mid }{\mathrm{\mid }}_1=2^{n-1}$ and it is easy to show that each $x_i$ must be even. Verifying a special case of a conjecture of Balister, Győri, and Schelp, we show that these conditions are also sufficient.