Holik–Kara's strengthened Yu conjecture

Let $F$ and $G$ be algebraically independent polynomials in $\mathbb{C}[x_1,\ldots,x_n]$. Assume that each generates its own centralizer in $\mathbb{C}[x_1,\ldots,x_n]$, that $\deg F\nmid\deg G$ and $\deg G\nmid\deg F$, that $F(0)=G(0)=0$, and that the linear parts of $F$ and $G$ are linearly independent. Holik–Kara's conjecture. Under these assumptions, one should have

$$\deg[F,G] > \min\\{\deg F, \deg G\\},$$

or perhaps a weaker inequality of the form

$$\deg[F,G] > c\cdot\min\\{\deg F, \deg G\\}$$

for some constant $c$ independent of $F$ and $G$. This reformulation is motivated by the failure of Yu's original conjecture and by the observation that the counterexample's linear parts are dependent, whereas components of a common polynomial automorphism have linearly independent linear parts. Its resolution is not given in the supplied text.