Strong Topological Rigidity of Non-Compact Orientable Surfaces

We show that every orientable infinite-type surface is properly rigid as a consequence of a more general result. Namely, we prove that if a homotopy equivalence between any two non-compact orientable surfaces is a proper map, then it is properly homotopic to a homeomorphism, provided surfaces are neither the plane nor the punctured plane. Thus all non-compact orientable surfaces, except the plane and the punctured plane, are topologically rigid in a strong sense.
Comment: 42 pages, 9 figures. v3: incorporates the referee's comments, accepted in the Algebraic & Geometric Topology