Gauge-theoretic flatness conjecture for Fock fields

Let $(P,\Phi_0,\sigma)$ be a $G$-Fock bundle over $S$ equipped with a compatible hermitian structure $\rho$. Let $\eta\in\Omega^0(S,\mathfrak{g}_P^{\sigma,-\rho})$ be a hermitian endomorphism-valued function, and set $\Phi=e^{-\eta}\Phi_0e^{\eta}$. Let $d_A$ be the corresponding canonical connection and let $\Phi^{*}=-\rho(\Phi)$. Gauge-theoretic flatness conjecture. There exists a unique such $\eta$ for which $\Phi$ is positive and

$$\Phi+d_A+\Phi^{*}$$

is flat, equivalently

$$F(A)+[\Phi\wedge\Phi^{*}]=0.$$

This is an equivalent reformulation of the main flatness problem with fixed hermitian structure and varying gauge class of the Fock field; the source provides no resolution.