A recent paper proves the existence of homeomorphic but not diffeomorphic Mazur manifolds (see also examples of exotic pairs of contractible Stein manifolds).

Let's call them $M_1$ and $M_2$. If we glue $W= M_1 \cup_{\partial M_1=\partial M_2} M_2$, then we get a manifold homeomorphic to $S^4$ by Freedman's theorem. Doubling $M_1$ or $M_2$ yields the standard smooth $S^4$ by a theorem of Mazur. But I'm wondering if $W$ is diffeomorphic to $S^4$?

Actually, I don't know if twisting the double of an Akbulut cork can yield an exotic $S^4$? Any exotic pair of manifolds are related by twisting along a cork, so I guess I'm asking whether anything more is known when the complementary contractible manifolds are homeomorphic?