Definition: An exotic $\mathbb{R}^4$ is a smooth open 4-manifold $E\mathbb{R}^4$ that is homeomorphic to $\mathbb{R}^4$ but not diffeomorphic to it.
Definition: An exotic $E\mathbb{R}^4$ is called small if it can be smoothly embedded as an open subset of the standard $\mathbb{R}^4$. We can assume that $E\mathbb{R}^4$ is an open inside of $\mathbb{R}^4$.
Question: Is there exist a small exotic $E\mathbb{R}^4\subset \mathbb{R}^4$ which is invariant by the antipode map?
By antipode map I mean the map whose sends $x$ to $-x$.