Kaledin–Verbitsky harmonic metric conjecture for NHYM bundles

Let $(E,\nabla)$ be a bundle with a non-Hermitian Yang–Mills connection $\nabla$. A bundle is $\nabla$-stable if it is stable with respect to the connection $\nabla$, and a harmonic metric is a metric satisfying the harmonicity equations associated with the NHYM connection. Kaledin–Verbitsky's harmonic metric conjecture. There exists a harmonic metric $h$ on $E$ if and only if $E$ is a direct sum of $\nabla$-stable bundles. Also, if $E$ itself is $\nabla$-stable, then $h$ is unique up to a constant factor. This is proposed as an analogue of the Uhlenbeck–Yau theorem, motivated by the hyperkähler reduction description of the NHYM equations; the source does not indicate a resolution.