Homoclinic intersections of symplectic partially hyperbolic systems with 2D center

We study some generic properties of partially hyperbolic symplectic systems with 2D center. We prove that $C^r$ generically, every hyperbolic periodic point has a transverse homoclinic intersection for the maps close to a direct/skew product of an Anosov diffeomorphism with a map on $S^2$ or $\mathbb{T}^2$