Stabilization of heterodimensional cycles

We consider diffeomorphisms f with heteroclinic cycles associated with saddles P and Q of different indices. We say that a cycle of this type can be stabilized if there are diffeomorphisms close to f with a robust cycle associated with hyperbolic sets containing the continuations of P and Q. We focus on the case where the indices of these two saddles differ by one. We prove that, excluding one particular case (so-called twisted cycles that additionally satisfy some geometrical restrictions), all such cycles can be stabilized.