This question was prompted by my answer to this question.

An exotic $\mathbb{R}^4$ is a smooth manifold homeomorphic to $\mathbb{R}^4$ which is not diffeomorphic to $\mathbb{R}^4$ with its standard smooth structure. An exotic $\mathbb{R}^4$ is said to be *small* if it can be embedded in the standard $\mathbb{R}^4$ as an open subset. An exotic $\mathbb{R}^4$ which is not small is called *large*.

Does every large $\mathbb{R}^4$ embed in $\mathbb{R}^5$?

Freedman and Taylor showed there is a maximal exotic $\mathbb{R}^4$, into which all other $\mathbb{R}^4$'s can be smoothly embedded as open subsets. So it would be enough to show that this one embeds in $\mathbb{R}^5$.

By the Whitney Embedding Theorem, every large $\mathbb{R}^4$ embeds in $\mathbb{R}^8$, but I suspect one can do better.