Diffeomorphisms of manifolds with boundary

Question

If two compact manifolds have diffeomorphic interiors and diffeomorphic boundaries, are they then diffeomorphic? Is it true for surfaces? Some context: there seems to exist an example by Barden and Mazur of a nontrivial cobordism between some oriented manifold $M$ and $-M$ whose interior is trivial (diffeo to $(0,1)\times M$). This would be a counterexample, but I cannot find the reference anywhere.