Unipotent elements in representations of finite groups of Lie type

Let G be a finite quasi-simple group of Lie type of defining characteristic r > 2. Let H = h,G be a group with normal subgroup G, where h is a non-central r-element of H. Let φ be an irreducible representation of H non-trivial on G over an algebraically closed field of characteristic = r. We show that φ(h) has at least two distinct eigenvalues of multiplicity greater than 1, unless G is a central quotient of one of the following groups: SL(2, r), SL(2, 9) or Sp(4, 3), and H = G · Z(H).