The inner-polynomial lattice-point conjecture

Let $\mathcal{Q}$ and $R_{m,n}$ be as above, and let $\zeta(F)$ denote the inner polynomial of $F$. The inner-polynomial lattice-point 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 is a lattice point $(m',n')$ such that $\zeta(F)\in R_{m',n'}$ and

$$(m',n')\in\left\{(x,y)\in\mathbb{R}^2:0\leq y<\frac{a-1}{a}n,\quad y=\frac{n}{m}x\right\}.$$

The paper describes this as a seemingly weaker conjecture intended to support the Magnus-formula approach; no resolution is supplied.