Matchings in the hypercube with specified edges

Given a matching M in the hypercube Qⁿ, the profile of M is the vector x= (x₁, , xₙ) Nⁿ such that M contains xᵢ edges whose endpoints differ in the ith coordinate. If M is a perfect matching, then it is clear that ||x||₁ = 2^n-1 and it is easy to show that each xᵢ must be even. Verifying a special case of a conjecture of Balister, Gyori, and Schelp, we show that these conditions are also sufficient.