  Let M be an orientable and irreducible 3-manifold whose boundary is an
incompressible torus. Suppose that M does not contain any closed nonperipheral
embedded incompressible surfaces. We will show in this paper that the immersed
surfaces in M with the 4-plane property can realize only finitely many boundary
slopes. Moreover, we will show that only finitely many Dehn fillings of M can
yield 3-manifolds with nonpositive cubings. This gives the first examples of
hyperbolic 3-manifolds that cannot admit any nonpositive cubings.
