Small-covector flatness conjecture for G-Fock bundles

Let $(P,\Phi)$ be a $G$-Fock bundle over $S$ with hermitian structure $\rho$ such that $\Phi\in\mathcal{P}$ is positive. Let $\mu$ be the induced $\mathfrak{g}$-complex structure on $S$, and let $t\in(Z(\Phi)\bar K)^*$ be a covector. The covector $t$ is $\mu$-holomorphic when the associated connection has curvature in $\operatorname{Im}(\operatorname{ad}_\Phi)$. Small-covector flatness conjecture. If $t$ is $\mu$-holomorphic and small, then there exists a gauge transformation $\eta\in\Omega^0(S,\mathfrak{g}_P^{-\rho})$ such that the 3-term connection associated to $e^{-\eta}\Phi e^{\eta}$ and $t$ is flat. This extends the main conjecture by incorporating a covector and asserts flatness for sufficiently small $\mu$-holomorphic deformations; the source gives no resolution status.