Low-degree conjecture for color factors of non-oriented vacuum graphs

Let $G$ be a non-oriented vacuum graph with $v_G\geq4$, and let its color factor be a polynomial or Laurent polynomial in $N$ whose low degree is the smallest exponent of $N$ with nonzero coefficient. Low-degree conjecture. The color factor of $G$ has low degree $\geq2$. Because nonzero color factors for graphs with fewer than four vertices are powers of $N^2-1$, this conjecture would imply that $R_G(N)\in\mathbb{Z}[N]$. The paper also notes that, for a connected non-oriented graph with even $h_G$, the low degree is odd; the conjectured bound in that case is at least $3$.