Let $M,N\subset \mathbb{R}^n$ be two open semi-algebraic subsets, and assume that $M$ and $N$ are $C^\infty$ diffeomorphic, i.e. isomorphic as smooth submanifolds of $\mathbb{R}^n$. Does this imply that $M$ and $N$ are also $C^\omega$-Nash diffeomorphic?
For the cases $n=1$ and $n=2$ the answer is positive - this follows immediately from Corollary 3 in Shiota's 1983 paper Classification of Nash manifolds. I think the answer in general may be positive, and this may somehow follow from this paper, or from some (other) result in Shiota's book "Nash Manifolds". For instance, by Corollary II.4.4 in this book it is enough to show that $M$ and $N$ are $C^r$-Nash diffeomorphic for some $r>0$.
My intuition is somehow supported by Theorem 2 in the mentioned paper (or Theorem VI.2.2 in the mentioned book) - loosely speaking it "says" that the only problem may come from the boundary, and for open subsets of $\mathbb{R}^n$ the boundary cannot be too bad.
