Harmonic metric conjecture for non-Hermitian Yang–Mills connections

Let $M$ be a compact Kähler manifold and let $\langle {\cal B},\nabla\rangle$ be a bundle with a non-Hermitian Yang–Mills $(1,1)$-connection. A subsheaf is called destabilizing when its slope is at least the slope of ${\cal B}$, and $\langle {\cal B},\nabla\rangle$ is called $\nabla$-stable when it has no destabilizing subsheaves. A Hermitian metric $h$ is harmonic when $\Lambda\Xi=0$, where $\Xi$ is the pseudocurvature of $\langle {\cal B},\nabla,h\rangle$.

Harmonic metric conjecture. There exists a harmonic metric $h$ on ${\cal B}$ if and only if ${\cal B}$ is a direct sum of $\nabla$-stable bundles. If ${\cal B}$ itself is $\nabla$-stable, then $h$ is unique up to a constant factor.

This is a non-Hermitian Yang–Mills analogue of the correspondence between stability and harmonic metrics for flat bundles. The supplied text gives no resolution status.