Dominating surface-group representations into ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ in the relative representation variety

Let $\rho$ be a representation of the fundamental group of a punctured surface into ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ that is not Fuchsian. We prove that there exists a Fuchsian representation that strictly dominates $\rho$ in the simple length spectrum, and preserves the boundary lengths. This extends a result of Gueritaud-Kassel-Wolff to the case of $$\mathrm{P}\mathrm{S}\mathrm{L}$_2 ($\mathbb{C}$)$-representations. Our proof involves straightening the pleated plane in ${\mathbb{H}}^3$ determined by the Fock-Goncharov coordinates of a framed representation, and applying strip-deformations.