Luo–Wu's discrete uniformization conjecture for hyperbolic convex hulls

Let $\mathbb{H}^3_P$ be the Poincaré ball model of hyperbolic $3$-space, let $\partial \mathbb{H}^3_P$ be its ideal boundary, and let $C_P(X)$ denote the hyperbolic convex hull of $X$. A compact subset $X\subset\partial \mathbb{H}^3_P$ is circle-type if every connected component of $X$ is either a round disk or a point. A genus zero complete hyperbolic surface has genus zero and a complete hyperbolic metric; an end is a cusp end when it is a cusp of that metric.

Discrete uniformization conjecture. Suppose that $S$ is a genus zero complete hyperbolic surface with countably many ends, all but at most one of which are cusp ends. Then $S$ is isometric to the boundary $\partial C_P(X)$ of the convex hull of some circle-type closed set $X\subset\partial \mathbb{H}^3_P$. Furthermore, such a circle-type closed set $X$ is unique up to Möbius transformations.

The statement is presented as a formulation of the discrete uniformization problem for non-compact polyhedral surfaces. In the supplied text, the broader countable-ends version is already proved, so this restricted statement is solved as a consequence rather than remaining open.