Uniqueness conjecture for fundamental families of an admissible Kleinian group

Let a Kleinian group be called admissible if it has generators $\varphi_i$, $i\in I_+$, satisfying the conditions of the amplification. For an admissible Kleinian group, a fundamental family is a family of domains $\widetilde{K}_{\bullet}$ satisfying those conditions. Two such families are equivalent if, after repeatedly applying inclusions $\widetilde{K}_{i}^{\left(1\right)}\subset\widetilde{K}_{i}^{\left(2\right)}$ for all $i\in I$, they are related by transitivity. Uniqueness conjecture. Any two fundamental families for an admissible Kleinian group are equivalent. Equivalent families already lead to the same spaces of sections of bundles; the conjecture would show that the curve with a fixed family of $A$-cycles is completely described by its corresponding Kleinian group.