Holonomy map fibers of $\mathbb{C}{\mathrm{P}}^1$-structures in moduli space

Let $S$ be a closed oriented surface of genus $g\ge 2$. Fix an arbitrary non-elementary representation $\rho \text{\hspace{0.17em}⁣}:{\pi }_1(S)\to {\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ and consider all marked (complex) projective structures on $S$ with holonomy $\rho$. We show that their underlying conformal structures are dense in the moduli space of $S$.