The 3-sum formula for Speyer's invariant

Let $G_1$ and $G_2$ be two $3$-connected graphs, each with a triangle $T_i\subset G_i$, and let $G_1\oplus_3G_2$ be a corresponding 3-sum. Define

$$p_i=\left|\pi_0\left(G_i\setminus V(T_i)\right)\right|$$

to be the number of connected components remaining after deleting the triangle vertices.

3-sum conjecture. The invariant satisfies

$$\mathsf{N}_2(G_1\oplus_3G_2)=\mathsf{N}_2(G_1)+\mathsf{N}_2(G_2)-p_1p_2.$$

The formula would reduce computations of $\mathsf{N}_2$ to highly connected graphs and includes a star--triangle relation as a special case. The paper reports computational confirmation but no proof.