Commuting pairs of endomorphisms of P^2

We consider commuting pairs of holomorphic endomorphisms of P^2 with disjoint sequence of iterates. The remaining case to be studied is when their degrees coincide after some number of iterations. We show in this case that they are either commuting Latt`es maps or commuting homogeneous polynomial maps of C^2 inducing a Latt`es map on the hyperplane at infinity.