Structural conjecture for planar graph complexes

Let $G$ be a connected planar graph. A cube complex is a chain complex

$$C=\bigoplus_{v\in\{0,1\}^n}C_v,$$

indexed by the vertices of an $n$-dimensional hypercube. For a vertex $v$, write $|v|=\sum_i v_i$ and let $\operatorname{St}_v(C)$ be its star subcomplex. Let $\langle G\rangle$ denote the complex associated with $G$, and let $\sigma$ be a handle map. Structural conjecture. There exists a cube complex $C=\bigoplus_{v\in\{0,1\}^n}C_v$ such that

$$\langle G\rangle\simeq\bigoplus_{v\in\{0,1\}^n}\left(\frac{t^2q^3}{1-t^2q^4}\right)^{|v|}\cdot\operatorname{Cone}^{|v|}\Big(\operatorname{St}_v(C)\xrightarrow{\sigma}q^2\operatorname{St}_v(C)\Big).$$

This is intended to describe a uniform structure for the complex associated with every connected planar graph. The source gives no evidence of a resolution, so the conjecture remains open.