Uniqueness conjecture for circle-type convex-hull realizations

Let $X,Y\subset\mathbb{C}$ be circle type closed sets, meaning that every connected component of each set is either a point or a round disk. Consider the boundaries of their convex hulls in $\mathbb{H}^3$.

Circle-type convex-hull uniqueness conjecture. If these two boundaries are isometric, then $X$ and $Y$ differ by a M\f6bius transformation.

Together with the corresponding existence conjecture, this would give uniqueness of the convex-hull realization within the class of circle type closed sets. The source reports the conjecture in finite-area and finite-ended cases, while general uniqueness remains open.