Plamenevskaya-type invariant survival conjecture for bridgeless plane graphs

Let $\Gamma$ be a bridgeless plane graph of a connected graph $G$. Let $\psi(\Gamma)$ be the invariant represented by the all-$x^{n-1}$ tensor in the $n$-color homology, and let $k$ be the quantum grading appearing in $CH_n^{0,k}(\Gamma;\mathbb{C})$. Invariant survival conjecture. The class

$$\psi(\Gamma)\in CH_n^{0,k}(\Gamma;\mathbb{C})$$

lives to the $E_\infty$-page of the spectral sequence. The conjecture is motivated by computations for bridgeless plane graphs and would provide a non-computer-based approach to the four-color theorem; whether the class always survives remains open.