The cubic graph relation between Speyer and Tutte polynomials

Let $G$ be a connected cubic graph with $n$ vertices. Write the Tutte polynomial as

$$\mathsf{T}_G(x,y)=\sum_{i,j}t_{i,j}(G)x^iy^j.$$

Cubic Tutte relation conjecture. The second derivative of Speyer's polynomial satisfies

$$g_G”(0)=2n\,t_{0,1}(G)-4t_{0,2}(G).$$

This would give a uniform relation between a coefficient of Speyer's polynomial and the first two relevant Tutte coefficients for connected cubic graphs. The paper reports data supporting the identity but does not prove it.