Intersection of holonomy varieties of $CP^1$-structures

Let $Σ$ be a closed orientable surface of genus at least two, and let $X, Y$ be distinct marked Riemann surface structures on $Σ$, possibly with opposite orientations. In this paper, we show that there are (exactly) countably infinite pairs of ${\rm CP}^1$-structures on $X$ and on $Y$ sharing holonomy $π_1(Σ) \to {\rm PSL}_2 {\rm C}$.