Given some simple normal crossings divisor $D$ on a complex manifold $X$, which is not assumed to be compact. Given a form $\omega\in H^0(X,\Omega_X^1(\log D))$, when is it possible to find a compactification $X\to \tilde X$ of $X$, where $\tilde D:=(\tilde X\setminus X)\cup D$ is a simple normal crossings divisor, such that $\omega$ extends to a global section of $\Omega_{\tilde X}^1(\log \tilde D)$?
A related question, is the extendibility of $\omega$ maybe independent of the compactification?
As an example of the kind of thing I'm looking for, we can note that a necessary condition is that the residues along the component of $D$ are constant, since otherwise they cannot extend to global functions on the compactifications of the components.