The structured Newton-polygon conjecture for Jacobian pairs

Let $\mathcal{R}=\mathbb{C}[x,y]$. For positive integers $m,n$, let $R_{m,n}$ be the set of polynomials whose Newton polygon is $\operatorname{Rect}_{m,n}$ and whose coefficient of $x^my^n$ is $1$. Let

$$\mathcal{Q}=\{(a,b,m,n)\in\mathbb{Z}_{>0}^4:a\mid m,\ a\mid n,\ \gcd(a,b)=1,\ 2\leq a<b\}.$$

The structured Newton-polygon conjecture. Suppose $(a,b,m,n)\in\mathcal{Q}$ and $F,G\in\mathcal{R}$ satisfy $[F,G]\in\mathbb{C}$, $F\in R_{m,n}$, and $G\in R_{bm/a,bn/a}$. Then $[F,G]=0$. This is introduced as a stronger structured form that implies the preceding conjecture and hence the Jacobian conjecture; no resolution is supplied.