Holomorphic continuation conjecture for stationary-measure solutions in the Schottky case

Let $\mu$ be a countably supported measure whose support generates a Fuchsian group $\Gamma \subset PSU(1,1)$ of second type. Let a solution be a function satisfying the equation referred to in the source as main equation, simp. Holomorphic continuation conjecture. Every such solution is holomorphic on

$$\overline{\mathbb{C}} \setminus \Lambda(\Gamma).$$

This statement is presented as out of reach for the moment in the supplied text, in the context of the Schottky case and the unknown behavior of the Cauchy transform of the Patterson–Sullivan measure.