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# Non-bimeromorphic compactifications

Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify this subset with $X$) and such that $\overline X\setminus X$ is a proper analytic subset (with no conditions on its codimension).

In particular, given two (different) compactifications of $X$, they always contain biholomorphic dense open subsets.

My question is: given two compactifications of $X$, are they necessarily bimeromorphic?

More precisely, does the biholomorphism between the two dense open subsets always extend to a global bimeromorphic map?

I guess the answer is no, but after a moment of reflection I cannot find any counterexample. Perhaps, it would suffice to look at compactifications of $\mathbb C^2$...