Hitchin's non-discrete image conjecture for branched conformal structures

Let $X$ be a Riemann surface of genus $g$, let $D$ be an effective divisor of degree $d$, and let $\mathfrak{U}\colon \mathfrak{S}^d\to \mathsf{Euler}^{-1}(2-2g+d)\subset\mathsf{Hom}(\pi,G)/G$ be the uniformization map from the moduli space of branched conformal structures. Hitchin's conjecture. Every representation with non-discrete image lies in the image of $\mathfrak{U}$. The map is a homotopy equivalence but is not surjective in general; the conjecture asserts that its failure to be surjective is confined to representations with discrete image.