Circle-type convex-hull realization conjecture for genus-zero hyperbolic surfaces

A circle type closed set in the Riemann sphere is a closed set whose connected components are each either a point or a round disk. Let $\Omega$ be a connected complete hyperbolic surface of genus zero.

Circle-type realization conjecture. There exists a circle type closed set $X\subset\mathbb{C}\cup\{\infty\}$ such that $\Omega$ is isometric to the boundary of the convex hull of $X$ in $\mathbb{H}^3$.

This conjecture concerns existence of convex-hull realizations for non-compact genus-zero hyperbolic surfaces. The source reports it in the finite-area, finite-ended, and countably-ended cases, but leaves the general case open.