Character-variety conjecture for the double Hamiltonian reduction

Let $\mathcal{A}//\mathcal{P}$ denote the space of parabolic connections with its infinitesimal higher-diffeomorphism action, and let $\operatorname{Symp}_0$ denote the corresponding group when the action is defined. Character-variety conjecture. The infinitesimal action on the space of parabolic connections can be integrated to an action of the group of higher diffeomorphisms, or a deformation of this group. The double Hamiltonian reduction

$$(\mathcal{A}//\mathcal{P})//\operatorname{Symp}_0$$

is an open dense subset of the character variety. This would identify the proposed double Hamiltonian reduction with a large part of the character variety, contingent on integrating the infinitesimal action or replacing the group by a deformation.