Homotopy conjecture for families of cuts on a complex curve

Let $M$ be a complex curve. Let $\gamma_{i}^{\left(1\right)}$, $i\in I^{\left(1\right)}$, be a disjoint family of smooth embedded cycles in $M$ such that $M\smallsetminus\bigcup\gamma_i$ is conformally equivalent to a fundamental domain of an admissible Kleinian group. Let $\gamma_{i}^{\left(2\right)}$ be another family satisfying the same conditions. Homotopy conjecture. All but finitely many cycles $\gamma_{i}^{\left(1\right)}$ are homotopic to cycles in $\gamma_{i}^{\left(2\right)}$. This is a geometric consequence proposed for the preceding uniqueness conjecture and concerns the extent to which different admissible systems of cuts describe the same curve.