The Fano criterion for affine canonical extensions with nef tangent bundle

Let $M$ be a compact Kähler manifold with nef tangent bundle $T_M$, and let $Z_M$ be its canonical extension associated with a Kähler class. Fano criterion conjecture. The canonical extension $Z_M$ is affine if and only if $M$ is Fano. The assertion is affirmed positively for surfaces by the paper's classification result, but is open for general compact Kähler manifolds.