Suppose $M$ is a smooth 4-manifold, and $U,V \subset M$ are exotic $\mathbb{R}^4$'s, i.e. homeomorphic to standard $\mathbb{R} ^4$. Further more suppose $U$ an $V$ intersect nicely sucht that $U \cup V \subset M$ is again homeomorphic to $\mathbb{R}^4$.

An exotic $\mathbb{R}^4$ is called small if it embeds into $S^4$.

Now my question:

If $U$ and $V$ are both small, is $U \cup V $ then also small? I thought about that for quite a long time, somehow I believe it should be true, but do not see any way to prove it.