Intersection of holonomy varieties of $C{P}^1$-structures

Let $\mathrm{\Sigma }$ be a closed orientable surface of genus at least two, and let $X,Y$ be distinct marked Riemann surface structures on $\mathrm{\Sigma }$, possibly with opposite orientations. In this paper, we show that there are (exactly) countably infinite pairs of ${\mathrm{C}\mathrm{P}}^1$-structures on $X$ and on $Y$ sharing holonomy ${\pi }_1(\mathrm{\Sigma })\to {\mathrm{P}\mathrm{S}\mathrm{L}}_2\mathrm{C}$.