The Newton-polygon divisibility conjecture for Jacobian pairs

Let $\mathcal{R}=\mathbb{C}[x,y]$, and let $N^0(F)$ denote the Newton polygon of $F$ with the origin as reference point. Let $a,b\in\mathbb{Z}_{>0}$ be relatively prime with $2\leq a<b$. Two polygons are similar with the origin as center when one is obtained from the other by a dilation about the origin. The Newton-polygon conjecture. Suppose that $F,G\in\mathcal{R}$ satisfy $[F,G]\in\mathbb{C}$ and that

$$\{(0,0),(0,1),(1,0)\}\subseteq N^0(F),$$

while $N^0(F)$ is similar to $N^0(G)$ with ratio $\deg(F):\deg(G)=a:b$. Then $[F,G]=0$. This is presented as a conjecture equivalent to the two-dimensional Jacobian conjecture, and the parser supplies no evidence of resolution.