Uniqueness conjecture for Schottky projective structures

Fix a marking of the projective structure $\Omega/\Gamma$ on $S$ with Schottky holonomy $\rho$. Let $\mathcal{P}_\rho$ be the collection of projective structures on $S$ with holonomy $\rho$ and the same orientation as $\Omega/\Gamma$. Let $Stab_\rho$ be the subgroup of orientation-preserving mapping classes $\phi$ satisfying $\rho\circ\phi^*=\rho$, and let $\mathcal{AML}_\rho(S)$ be the isotopy classes of multiloops consisting of disjoint admissible loops on $\Omega/\Gamma$. For a mapping class $\phi$, write $\operatorname{Supp}(\phi)$ for its minimal subsurface of nontrivial action. Uniqueness conjecture. Every $C\in\mathcal{P}_\rho$ can be obtained by changing the marking of $\Omega/\Gamma$ by a unique $\phi\in Stab_\rho$ and grafting $\Omega/\Gamma$ along a unique $L\in\mathcal{AML}_\rho(S)$ such that $L$ and $\operatorname{Supp}(\phi)$ are disjoint; equivalently,

$$\mathcal{P}_{\rho}(S)\cong\{(\phi,L)\in Stab_{\rho}\times\mathcal{AML}_{\rho}(S)\mid \operatorname{Supp}(\phi)\cap L=\emptyset\}.$$

Theorem supplies such a pair without guaranteeing disjointness, so the conjecture asserts that the intersection can always be uniquely resolved. The subgroup $Stab_\rho$ is known to be generated by Dehn twists along loops in $\ker(\rho)$, but the asserted uniqueness with disjoint support remains open.