Apostolov–Calderbank–Gauduchon–Tønnesen-Friedman projective-bundle conjecture

Let $(X,J)=P(E)$ be a projective bundle over a compact curve $\Sigma$ of genus at least $2$. A subbundle is stable in the usual slope-stability sense. Apostolov–Calderbank–Gauduchon–Tønnesen-Friedman conjecture. The projective bundle $(X,J)=P(E)$ admits an extremal metric in some Kähler class if and only if

$$E=\bigoplus_{i=0}^l E_i,$$

where $E_i$ is a stable subbundle for $1\leq i\leq l$. This conjecture characterizes the existence of extremal metrics on these projective bundles through a decomposition of the underlying bundle. The paper states that, assuming uniformly bounded Calabi-flow curvature, its method provides a proof; the supplied material does not give further details of that proof.