The orbit-equivalence conjecture for prequantum implicit representations

Let $\mathcal{O}$ be a $\operatorname{Diff}_0(M)$-orbit in the explicit shape space $\mathcal{X}$, let $\mathcal{F}_{\mathcal{O}}\coloneqq \Pi^{-1}\mathcal{O}$ be the entire fiber bundle over $\mathcal{O}$, and let $\mathcal{U}$ be a $\operatorname{DC}$-orbit in $\mathcal{F}_{\mathcal{O}}$. Suppose each connected component of $M$ intersects $\operatorname{im}\gamma$ for $\gamma\in\mathcal{O}$. Orbit-equivalence conjecture. Then

$$\mathcal{F}_{\mathcal{O}}=\mathcal{U}.$$

The paper presents this as the expected higher-dimensional extension of the equality established in the base-dimension-two case, asserting that the full fiber bundle is a single orbit under the combined diffeomorphism and exponential-function action.