This is closely related to the question here. The setup is that $U\subset\mathbb{C}$ is an open bounded simply connected domain with $C^\infty$ boundary. If $\phi:U\rightarrow\mathbb{D}$ is a biholomorphic mapping from $U$ to the unit disk, it is known that $\phi$ extends to a smooth map $\overline{U}\rightarrow\overline{\mathbb{D}}$. However, the question that is not addressed is whether this extension is actually a diffeomorphism. Now a comment by the OP on the linked question states that Graeme Segal showed something like this in some unpublished notes, which I can't find anywhere. So let me now ask this precisely:

Does $\phi$ as above extend to a diffeomorphism? The answer seems to be yes, but I would appreciate any reference that confirms this.

Incidentally, I'm also interested in analogous $C^k$-boundary and boundary-with-corners statements, but those seem even more exotic.