The physicists' conjecture on global Coulomb gauge fixing

Let $M$ be the compact 3-manifold with boundary under consideration, let $\rho:P\to M$ be the principal gauge bundle, let $\mathcal{C}$ be its space of connections, and let $\mathcal{G}$ be the gauge group. For a connection $\nabla_A\in\mathcal{C}$, write

$$H_A=\{\tau: \tau\text{ is a $\mathfrak{k}$-valued 1-form and }d_A^*\tau=0\}$$

and

$$\mathcal{S}_A=\{A+\tau:\tau\in H_A\}\subseteq\mathcal{C}.$$

Physicists' conjecture. Fix a connection $\nabla_A\in\mathcal{C}$. Then for every $\nabla_{A'}\in\mathcal{C}$, there exists a unique $g\in\mathcal{G}$ such that $\nabla_{A'}\cdot g\in\mathcal{S}_A$.

This asserts that every gauge orbit meets the Coulomb slice based at $A$ exactly once, providing the global section physicists seek in order to remove the Gribov ambiguity and define the functional integral over the quotient. The supplied excerpt gives no evidence that the conjecture has been proved or disproved.