Louboutin's degree conjecture for the auxiliary Laurent polynomial

Let $a,b\in\mathbb{Z}$ with $a\neq 0$ and $b\geq 1$, and let $m_{a,b}:=a^2+ab+b^2$. Using the polynomials and Laurent polynomials defined in Proposition 19, let $G_{a,b,m}(T)$ and $B_{a,b,m}$ be the associated quantities, with $M_{a,b}$ and $N_{a,b,m_{a,b}}$ taken from the cited table. Louboutin's degree conjecture. If $m_{a,b}$ is odd and at least $5$, and $(a,b)$ is not of the excluded forms $(-2b,b)$ with odd $b$, $(b,b)$ with odd $b$, or $(b,b)$ with even $b\geq 2$, then

$$\deg G_{a,b,m_{a,b}}(T)<0\qquad\text{and}\qquad B_{a,b,m_{a,b}}\leq m_{a,b}.$$

These inequalities assert the hypotheses needed in Proposition 19 and hence support the paper's main theorem; the supplied text does not specify whether the conjecture was already resolved.