Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By Nagata's theorem, there exists a proper binational map $X'\to X$ that is an isomorphism on $U$ such that $\phi$ extends to $X'\to Y$.

In characteristic zero, by Hironaka's theorem we can choose $X'\to X$ to be a sequence of blow-ups in smooth centers. Is anything known in this direction in positive characteristics, without modifying $U$ with alterations? Is the problem easier if the target $Y$ is a curve or a surface?