Extremal metric decomposition conjecture for projective bundles over curves

Let $(M,J)=P(E)$ be a projective bundle over a compact curve of genus at least $2$, where $E$ is a holomorphic vector bundle. A bundle decomposes as a direct sum of stable subbundles when $E=\bigoplus_i E_i$ with each $E_i$ stable.

Extremal metric decomposition conjecture. The projective bundle $(M,J)=P(E)$ admits an extremal Kähler metric in some Kähler class if and only if $E$ decomposes as a direct sum of stable subbundles.

This would give a general existence criterion for extremal Kähler metrics on projective bundles over curves. The paper presents it as a statement suggested by the compatibility conjecture; its resolution is not supplied in the given text.