Greb–Wong conjecture on Stein and affine canonical extensions

Let $M$ be a compact Kähler manifold, and let $\alpha\in H^1(M,\Omega_M)$ be a Kähler class. Consider the extension

$$0\to\mathcal O_M\to V^\alpha\to T_M\to 0$$

and the associated canonical extension

$$Z_M:=\mathbb P(V^\alpha)\setminus\mathbb P(T_M).$$

Here, $T_M$ is the tangent bundle of $M$, and a vector bundle is nef or big in the usual sense; a complex manifold is called affine if it is biholomorphic to the analytification of an affine variety. Greb–Wong's conjecture. If $Z_M$ is a canonical extension defined by some Kähler class on $M$, then $Z_M$ is Stein if and only if $T_M$ is nef, and $Z_M$ is affine if and only if $T_M$ is nef and big. This conjecture predicts precise positivity criteria for the Stein and affine properties of canonical extensions. The affine implication from nefness and bigness is proved in the paper, while the converse is related to a conjecture of Goodman and remains open.