Decomposing the cube into paths

We consider the question of when the nn-dimensional hypercube can be decomposed into paths of length kk. For odd nn it is necessary that kk divides n2n−1n2n−1 and that k≤nk≤n. Anick and Ramras (2013) conjectured that these two conditions are also sufficient for all odd nn and prove that this is true for odd n≤232n≤232. In this note we prove the conjecture.