Uniformization conjecture for combinatorial Ricci flow

Let $\mathcal{T}$ be an infinite disk triangulation with intersection angles $\Phi\in [0,\frac{\pi}{2}]^E$. Its $1$-skeleton is the graph formed by its vertices and edges, and a graph is VEL-hyperbolic when it has the corresponding vertex extremal-length property. A locally finite univalent circle packing of $\mathcal{T}$ in the unit disc is a circle packing whose contact graph is $\mathcal{T}$ and whose circles are locally finite and univalent.

Uniformization of the CRF. 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 univalent circle packing of $\mathcal{T}$ in the unit disc; and the $1$-skeleton of $\mathcal{T}$ is VEL-hyperbolic.

The conjecture extends the equivalence between convergence of combinatorial Ricci flow, existence of a hyperbolic circle-packing metric, and VEL-hyperbolicity from the finite setting to infinite disk triangulations. The source notes that the second and third conditions are equivalent by He’s theorem and that convergence implies the existence of a hyperbolic circle-packing metric; the implication from VEL-hyperbolicity to convergence remains to be proved.