Planar graph moduli and unlabeled configuration space conjecture

Let $\mathcal N_n(\mathbb R^2)$ be the moduli space of planar graphs with $n$ bounded faces, and let $\operatorname{uConf}_n(\mathbb R^2)$ be the configuration space of $n$ unlabeled points in the plane. Planar graph configuration-space conjecture. There is a homotopy equivalence

$$\mathcal N_n(\mathbb R^2)\simeq \operatorname{uConf}_n(\mathbb R^2).$$

The source describes this as a simpler two-dimensional combinatorial parallel and says that no proof was found in the literature.