Canonical extensions of manifolds with nef tangent bundle

To any compact Kähler manifold $(X,\omega )$ one may associate a bundle of affine spaces $Z_X\to X$ called a \emph{canonical extension} of $X$. In this paper we prove that if the tangent bundle of $X$ is nef, then the total space $Z_X$ is a Stein manifold. This partially answers a question raised by Greb-Wong of whether these two properties are actually equivalent. We also complement some known results for surfaces in the converse direction.