The asymptotic behavior of the explicit solution to the Bene filtering problem is studied. It is shown that there is a universal, data-dependent change of location that renders any Bene filter asymptotic to a fixed normal distribution. Asymptotic stability of Bene filters follows as a result; that is, the variational distance between any two, differently initialized solutions of the Kushner-Stratonovich equation converges to zero in the infinite time limit. It is also shown the relative entropy between differently initialized solutions converges to zero. More careful relative entropy bounds are used to derive exponential convergence of the variational distance between filters