The Feynman-period characterization of Speyer's invariant

A graph is cyclically $6$-connected if it is $4$-regular and has no $4$-edge cut other than one isolating a vertex; cyclically $4$-connected $3$-regular graphs are defined analogously. For such a graph $G$, let $\widetilde{\mathcal{P}}(G)$ denote its Feynman period after deleting any vertex, and let $\mathsf{N}_2(G)$ be the second Speyer invariant.

Period--Speyer conjecture. If two cyclically $6$-connected $4$-regular graphs, or two cyclically $4$-connected $3$-regular graphs, satisfy

$$\widetilde{\mathcal{P}}(G_1)=\widetilde{\mathcal{P}}(G_2),$$

then

$$\mathsf{N}_2(G_1)=\mathsf{N}_2(G_2).$$

This would make $\mathsf{N}_2$ a function of the Feynman period on these graph classes. It is supported by all graphs with known periods but remains open.