Fuchsian singularity conjecture for hitting measures

Let $(X_n)$ be a finite-range admissible random walk generated by a probability measure $\mu$ on a cocompact Fuchsian group $\Gamma$, and let $\nu_\mu$ be its hitting measure on the boundary $\partial \Gamma$, identified with $S^1$. Fuchsian singularity conjecture. For every such random walk, the hitting measure $\nu_\mu$ is singular with respect to Lebesgue measure on

$$S^1 \simeq \partial \Gamma.$$

This is the singularity conjecture for cocompact Fuchsian groups. The paper studies this question through explicit asymptotic estimates for first-passage functions and proves the singularity of the hitting measure for a specific class of random walks, while the stated generality remains open.