Perfect matchings in rr-partite rr-graphs

Let HH be an rr-partite rr-graph, all of whose sides have the same size nn. Suppose that there exist two sides of HH, each satisfying the following condition: the degree of each legal r−1r−1-tuple contained in the complement of this side is strictly larger than n2. We prove that under this condition HH must have a perfect matching. This answers a question of Kühn and Osthus.