The Newton-polygon divisibility criterion for the two-dimensional Jacobian conjecture

Let $\mathcal{R}=\mathbb{C}[x,y]$, and let $N(F)$ denote the Newton polygon of $F$. Let $a,b\in\mathbb{Z}_{>0}$ be relatively prime with $2\leq a<b$. Suppose that $F,G\in\mathcal{R}$ satisfy $[F,G]\in\mathbb{C}$, that

$$\{(0,0),(0,1),(1,0)\}\subset N(F),$$

and that $N(F)$ is similar to $N(G)$ with the origin as center of similarity and ratio $\deg(F):\deg(G)=a:b$.

The Newton-polygon criterion. Then $[F,G]=0$.

The source states this as an equivalent reformulation of a divisibility criterion for the Jacobian conjecture, using the similarity of augmented Newton polygons for Jacobian pairs. It is presented as an open conjectural route to the Jacobian conjecture.