Demailly–Campana–Peternell conjecture on cotangent bundles with affine contraction

Let $M$ be a smooth algebraic variety, set $X=T^*M$, and let

$$X\longrightarrow Y=H^0(X,\operatorname{\cal O}_X)$$

be the natural map, assumed to be projective and birational. Demailly–Campana–Peternell conjecture. Then $M=G/P$, where $G$ is a semisimple algebraic group and $P\subset G$ is a parabolic subgroup.

This conjecture characterizes the smooth varieties whose cotangent bundles admit the specified projective birational contraction. It is described as difficult and longstanding; no resolution is supplied here.