Jie-Tai Yu's degree estimate for commutators

Let $K\langle X\rangle$ be a free associative algebra, and let $f,g\in K\langle X\rangle$ be algebraically independent polynomials whose homogeneous components of maximal degree are algebraically dependent. Suppose that $f$ and $g$ generate their centralizers $C(f)$ and $C(g)$, respectively.

Jie-Tai Yu's conjecture. If neither $\operatorname{deg}(f)$ nor $\operatorname{deg}(g)$ divides the other, then

$$\operatorname{deg}([f,g])>\min\{\operatorname{deg}(f),\operatorname{deg}(g)\}.$$

The divisibility and centralizer-generation hypotheses are necessary according to the examples discussed in the source. If true, the conjecture would yield a stronger description of the tame automorphism group of $K\langle x,y,z\rangle$ and address the lack of degree estimates for commutators relevant to recognizing wild automorphisms.