In this article about completeness of semi-riemannian manifolds the author writes (in section 2.2) that it is unkown if the following statement holds:

A compact indefinite manifold which is conformal to a complete one is complete.

The author also gives a proof for the weaker statement: Let $g'=e^{\sigma}g$ be two conformal metrics. Then $g'$ is null complete if and only if $g$ is null complete. So the above statement holds if one replaces "complete" by "null complete":

Since this article is from 1994, I'm asking myself if there has been made any progress in proving (or disproving) the above statement since then. I already looked through the publication list of the author and tried excessive googling. But I couldn't find anything.