Perfect matchings in r-partite r-graphs

Let H be an r-partite r-graph, all of whose sides have the same size n. Suppose that there exist two sides of H, each satisfying the following condition: the degree of each legal (r-1)-tuple contained in the complement of this side is strictly larger than n/2. We prove that under this condition H must have a perfect matching. This answers a question of Kuhn and Osthus.