Gallo–Kapovich–Marden grafting conjecture for complex projective structures

Let $\Sigma$ be an orientable surface with finitely many punctures and no boundary, and let $\mathcal{P}(\Sigma)$ denote its deformation space of complex projective structures. Each structure has a holonomy representation in $\mathcal{R}(\Sigma)$, the space of representations of $\pi_1(\Sigma)$ into $\mathrm{PSL}_2\mathbb{C}$ up to conjugation. Gallo–Kapovich–Marden grafting conjecture. Two complex projective structures have the same holonomy if and only if it is possible to obtain one from the other by some sequence of graftings and degraftings. This conjecture asks for a geometric description of all complex projective structures with the same holonomy; it remains unresolved in the generality stated here.