Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\hookrightarrow \bar{X}$ such that $\bar{X}\setminus X$ is a strict normal crossings divisor, is not known in positive characteristic.
Instead, even if $X$ is not regular, there is de Jong's theorem on alterations which says that there is a dominant proper generically finite morphism (this is called an alteration) $\phi:X'\rightarrow X$, with $X'$ regular such that $X'$ has a good compactification.
In a review of de Jong's paper, Oort writes (bottom of p.5, emphasis mine):
"Also, when starting with a singular X, it might be that the morphism $\phi : X' \rightarrow X$ thus constructed need not be finite above non-singular points of X."
Even though he doesn't explicitly claim it, this sentence seems to suggest that for non-singular $X$ the situation is different. Hence my question (even though I suspect the answer is 'no'):
If $X$ is non-singular, does there always exist a finite alteration $X'\rightarrow X$, such that $X'$ is regular and has a good compactification?
