The vanishing innermost-polynomial conjecture

Let $\mathcal{Q}$ and $R_{m,n}$ be as above. For $F\in R_{m,n}$, let $Z$ denote the innermost polynomial of $F$. The vanishing innermost-polynomial 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

$$Z=0.$$

The paper uses this assertion in its proposed chain of implications toward the Jacobian conjecture; no resolution is supplied.