The perfect Martin invariant conjecture for cyclically 6-connected 4-regular graphs

Let $G_1$ and $G_2$ be cyclically $6$-connected $4$-regular graphs, and let $v_1$ and $v_2$ be vertices whose deletion produces graphs with periods $\mathcal{P}(G_1\setminus v_1)$ and $\mathcal{P}(G_2\setminus v_2)$. Write $\operatorname{\mathsf{M}}(G^{\bullet})$ for the Martin sequence. Perfect Martin invariant conjecture.

$$\mathcal{P}(G_1\setminus v_1)=\mathcal{P}(G_2\setminus v_2)\quad\Longleftrightarrow\quad \operatorname{\mathsf{M}}(G_1^{\bullet})=\operatorname{\mathsf{M}}(G_2^{\bullet}).$$

The claim is motivated by numerical correlations and computations comparing known periods, Hepp bounds, and Martin sequences; its resolution for this class is not supplied in the paper.