The 4-edge-twist congruence for Speyer's polynomial

A 4-edge twist of a graph $G_1$ is obtained from a minimal $4$-edge cut by reconnecting the four cut edges according to a double transposition. Let $G_2$ be any 4-edge twist of $G_1$.

4-edge-twist conjecture. Then

$$g_{G_1}(t)\equiv g_{G_2}(t)\pmod{t^3\mathbb{Z}[t]}.$$

Thus a 4-edge twist would preserve the first two nonzero coefficients of Speyer's polynomial. The paper presents this as a computationally supported relation and leaves it open.