Planarity conjecture for alternate-generator origami orbit graphs

Let $n$ be a positive integer. Using the generating set $\{P,R\}$ for $\operatorname{SL}(2,\mathbb{Z})$, where

$$P=\begin{bmatrix}0&1\\-1&1\end{bmatrix},\qquad R=\begin{bmatrix}0&-1\\1&0\end{bmatrix},$$

let $\overline{\mathcal{G}}_{n}$, $\overline{\mathcal{G}}_{n}^{A}$, and $\overline{\mathcal{G}}_{n}^{B}$ be the associated orbit graphs. Planarity conjecture. The graphs $\overline{\mathcal{G}}_{n}$, $\overline{\mathcal{G}}_{n}^{A}$, and $\overline{\mathcal{G}}_{n}^{B}$ are planar for $n\leq 7$; $\overline{\mathcal{G}}_{9}^{B}$ is planar; and all remaining graphs are non-planar. The conjecture is motivated by the fact that this planarity range exactly agrees with the range for genus-zero components of the Weierstrass curves $W_{n^{2}}$ given by Mukamel. Planarity for $n\leq 7$ and for $\overline{\mathcal{G}}_{9}^{B}$ is computationally confirmed; the non-planarity of all remaining graphs remains conjectural.