Greb–Wong–Höring–Peternell conjecture on canonical extensions and tangent-bundle positivity

Let $X$ be a compact Kähler manifold. A canonical extension $Z_X$ is the canonical extension associated with a Kähler class on $X$.

Greb–Wong–Höring–Peternell conjecture. The tangent bundle $d4af_X$ is nef, respectively big and nef, if and only if some canonical extension $Z_X$ of $X$ is Stein, respectively affine.

This conjecture connects positivity of the tangent bundle with the analytic and algebraic properties of canonical extensions. The source attributes it to work of Greb–Wong and Höring–Peternell; no general resolution is stated.