Let $M$ be a compact connected three manifold. By an exotic pair of bounding 4-manifolds, I mean two smooth 4-manifolds $X_1,X_2$ such that $X_1$ and $X_2$ are homeomorphic but not diffeomorphic, and $\partial X_1$ and $\partial X_2$ are homeomorphic to $M$.
I imagine that all 3-manifolds have exotic pairs of bounding 4-manifolds. In fact, I would imagine that we could just take some exotic pair of closed 4-manifolds $W_1, W_2$ and any bounding 4-manifold $X$ for $M$ and then just take $X_1 = X \sharp W_1$ and $X_2 = X \sharp W_2$.
Does this process always produce an exotic pair for $M$?
Do all 3-manifolds have an exotic pair that they bound?