The planar 4-connected conjecture for Speyer's third invariant

Let $G$ be a planar $4$-connected graph, meaning that no set of at most three vertices disconnects it. Let $\mathsf{N}_3(G)$ denote the third coefficient invariant associated with Speyer's polynomial.

Planar 4-connectivity conjecture. Then

$$\mathsf{N}_3(G)=0.$$

The conjecture is supported by the paper's dataset of planar 3-connected graphs and would extend the observed planar identity for $\mathsf{N}_2$ to the next invariant in the 4-connected case.