The perfect Martin sequence conjecture for cyclically 4-connected 3-regular graphs

Let $G_1$ and $G_2$ be cyclically $4$-connected $3$-regular graphs with the same number of vertices, and let $v_1$ and $v_2$ be vertices whose deletion produces periods $\mathcal{P}(G_1\setminus v_1)$ and $\mathcal{P}(G_2\setminus v_2)$. Define the Martin sequence by

$$\operatorname{\mathsf{M}}(G^{\bullet})=\left(\operatorname{\mathsf{M}}(G^{[2]}),\operatorname{\mathsf{M}}(G^{[4]}),\operatorname{\mathsf{M}}(G^{[6]}),\ldots\right).$$

Perfect Martin sequence conjecture.

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

Available data is compatible with this purely combinatorial characterization of equal periods, but the general assertion remains unproved.