The infinite Koebe–Andreev–Thurston theorem for plane cellular decompositions

Let $G$ and $\hat{G}$ be a pair of simple, 3-connected plane graphs corresponding to a cellular decomposition of $\mathbb{C}$ and its dual. A circle configuration is a configuration of circles whose tangency graph is the associated plane graph; write these configurations as $B$ and $\hat{B}$, and denote their tangency graphs by $G_B$ and $G_{\hat{B}}$. Infinite Koebe–Andreev–Thurston conjecture. There exist circle configurations $B$ and $\hat{B}$ such that

$$G_B \cong G \quad\text{and}\quad G_{\hat{B}} \cong \hat{G},$$

and these configurations are unique up to similarity. The conjecture would extend the finite Koebe–Andreev–Thurston theorem and known infinite results from triangulations to general cellular decompositions of the plane. The source provides no resolution, so the claim remains open.