The generator-composition conjecture for Jacobian pairs

Let $\mathcal{Q}$ and $R_{m,n}$ be as above. For a polynomial $F\in R_{m,n}$, let $W_F$ be the $F$-generator constructed in the paper, and let $\mathbb{T}$ be the corresponding class of one-variable polynomials. The generator-composition conjecture. If $(a,b,m,n)\in\mathcal{Q}$ and $F,G\in\mathbb{C}[x,y]$ satisfy $[F,G]\in\mathbb{C}$, $F\in R_{m,n}$, and $G\in R_{bm/a,bn/a}$, then there exists $\alpha\in\mathbb{T}$ with $\deg\alpha\geq2$ such that

$$F=\alpha(W_F).$$

The paper states that this conjecture implies the structured Newton-polygon conjecture and therefore the Jacobian conjecture; no resolution is supplied.