Superinjective simplicial maps of the two-sided curve complexes on nonorientable surfaces

Let N be a compact, connected, nonorientable surface of genus g >= 5 with n >= 0 boundary components. Let T(N) be the two-sided curve complex of N. If lambda: T(N) -> T(N) is a superinjective simplicial map, then there exists a homeomorphism h: N -> N unique up to isotopy such that H(alpha) = lambda(alpha) for every vertex alpha in T(N) where H = [h].