The planar 3-connected graph conjecture for Speyer's polynomial

A graph $G$ is 3-connected if it remains connected after the deletion of any set of at most two vertices, and it is planar if it can be drawn in the plane without edge crossings. Let $4\mathsf{N}_2(G)4$ denote the coefficient of $t^2$ in Speyer's matroid polynomial $g_G(t)$. The planar 3-connected graph conjecture. If a $3$-connected graph $G$ is planar, then

$$\mathsf{N}_2(G)=1.$$

This proposes a uniform value for the second coefficient of Speyer's polynomial on planar 3-connected graphs; the paper presents it as a structural property suggested by computations, and its resolution is not given.