The 3-edge-cut congruence for Speyer's polynomial

Let $G$ be a $3$-vertex-connected graph with a $3$-edge cut $C$, so that deleting $C$ gives two connected components $S\sqcup T$. Let $A=G/T$ and $B=G/S$ be obtained by contracting one side of the cut, replacing it by a $3$-valent vertex.

3-edge-cut conjecture. The Speyer polynomials satisfy

$$g_G(t)\equiv\frac{g_A(t)g_B(t)}{t}\pmod{t^3\mathbb{Z}[t]}.$$

The congruence determines the linear and quadratic terms of $g_G(t)$ from the two contracted graphs. The paper reports confirmation for all tested cuts, while higher coefficients are explicitly shown not to be determined in general.