Proof of the singularity conjecture for discrete subgroups of ${\mathrm{S}\mathrm{L}}_2(\mathbb{R})$

Let $\mu$ be a finitely supported probability measure on ${\mathrm{S}\mathrm{L}}_2(\mathbb{R})$ whose support generates a Zariski-dense discrete subgroup. We prove that its stationary probability measure on ${\mathbb{P}}^1(\mathbb{R})$ is singular with respect to the $\mathrm{S}\mathrm{O}(2)$-invariant probability measure. The proof compares winding variances on finite covers of a closed hyperbolic surface and uses a topological obstruction.