Ideal circle-pattern uniformization conjecture for combinatorial Ricci flow

Let $\mathcal{T}=(V,E,F)$ be an infinite cellular decomposition of an open disk with intersection angles $\Phi\in (0,\pi)^E$. Its $1$-skeleton is the graph formed by its vertices and edges, and a locally finite ideal circle pattern in the unit disc is a locally finite family of ideal circles with contact graph $\mathcal{T}$. The Poincaré dual of $\mathcal{T}$ is the cellular decomposition whose vertices correspond to the faces of $\mathcal{T}$ and whose incidence relations are dual to those of $\mathcal{T}$. The carrier of an ideal polyhedron is the union of the discs on the sphere corresponding to its faces.

Ideal circle-pattern uniformization conjecture. The following statements are equivalent: the combinatorial Ricci flow on $\mathcal{T}$ converges for every initial value in hyperbolic background geometry; there exists a locally finite ideal circle pattern with contact graph $\mathcal{T}$ in the unit disc; the $1$-skeleton of $\mathcal{T}$ is VEL-hyperbolic; and there exists an infinite ideal polyhedron $\mathcal{P}$ in $\mathbb{H}^3$ combinatorially equivalent to the Poincaré dual of $\mathcal{T}$ whose carrier is an open hemisphere of $\mathbb{S}^2$.

This conjecture proposes an analogous uniformization theorem for ideal circle patterns on infinite cellular decompositions, linking Ricci-flow convergence, circle-pattern existence, vertex extremal-length hyperbolicity, and ideal polyhedra in hyperbolic three-space. The source presents the equivalence as a conjecture and gives no resolution.