Suppose I have two smooth complex affine varieties $X$ and $Y$. Assume that they are each diffeomorphic to $\mathbb{R}^{2n}$ (where $n\geq 3$).

Question: If there exists open dense subsets $U\subset X$ and $V\subset Y$ such that $U$ is biholomophic to $V$, does there exist an extension to a biholomorphism between $X$ and $Y$?

Given that Hartog's Extension Theorem seems to give an affirmative answer when $X\setminus U$ is compact, I suspect that answer is no since $X\setminus U$ is not necessarily compact in this setting.

Natural generalizations: More generally, one can remove the assumption that $X$ and $Y$ are diffeomorphic to affine space and simply ask that they are diffeomorphic. Also, one can weaken the hypothesis that $X$, $Y$ are complex affine varieties and assume instead they are Stein manifolds.

Motivation: Given that exotic affine spaces need not be biholomorphic (see this MO question, its answer, and the main reference), an affirmative answer here would imply that there exist exotic affine spaces that are not birational.

This would potentially answer this MO question if one can show that equality in the Grothendieck ring would imply birationallity in this context (as suggested in the comments).

Remark: Regardless of the motivation, this question, and its generalizations, seem to me might be of independent interest, so I decided to post it separately. However, as I am not an analyst, I worry that I am asking a question with a trivial counter-example. Apologies if that is the case.


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