Symplectic Structures on the cotangent bundles of open 4-manifolds

We show that, for any two orientable smooth open 4-manifolds $X_0,X_1$ which are homeomorphic, their cotangent bundles $T^*X_0,T^*X_1$ are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold $R$ homeomorphic to $\mathbb{R}^4$, the standard Stein structure on $T^*R$ is Stein homotopic to the standard Stein structure on $T^*\mathbb{R}^4 = \mathbb{R}^8$. We use this to show that any exotic $\mathbb{R}^4$ embeds in the standard symplectic $\mathbb{R}^8$ as a Lagrangian submanifold. As a corollary, we show that $\mathbb{R}^8$ has uncountably many smoothly distinct foliations by Lagrangian $\mathbb{R}^4$s with their standard smooth structure.