The Stein universal-cover conjecture for nef cotangent bundles

Let $Y$ be a projective manifold, and let its universal cover be a Stein manifold. The cotangent bundle of $Y$ is nef if every curve has nonnegative degree against it.

Stein universal-cover conjecture. If the universal cover of $Y$ is a Stein manifold, then the cotangent bundle of $Y$ is nef.

This conjecture connects Steinness of universal covers with positivity of cotangent bundles and is motivated by the Singer–Hopf and Shafarevich conjectures. Its resolution status is not specified in the source.