Main flatness conjecture for G-Fock bundles

Let $(P,\Phi,\sigma)$ be a $G$-Fock bundle. A hermitian structure $\rho$ is compatible when it commutes with $\sigma$, and positive when the associated positivity condition for the Fock field holds. For such $\rho$, let $d_A$ be the unique unitary, $\sigma$-invariant connection satisfying $d_A\Phi=0$, put $\Phi^{*}=-\rho(\Phi)$, and write $F(A)$ for the curvature of $d_A$. Main flatness conjecture. There exists a unique compatible positive hermitian structure $\rho$ such that

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

is flat, equivalently

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

This is the paper's main conjecture and is motivated by the nonabelian Hodge correspondence; the source presents evidence but no proof or resolution.