Drensky–Yu's Poisson-bracket degree conjecture

Let $k$ be a field of characteristic $0$, and let $f$ and $g$ be algebraically independent polynomials in $k[x_1,\ldots,x_n]$. Assume that the homogeneous components of maximal degree of $f$ and $g$ are algebraically dependent, that $f$ and $g$ generate their integral closures $C(f)$ and $C(g)$ in $k[x_1,\ldots,x_n]$, respectively, and that neither $\deg f\mid\deg g$ nor $\deg g\mid\deg f$. Drensky–Yu's conjecture. One has

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

The conjecture gives a proposed lower bound for the degree of the Poisson bracket of two polynomials and is used in the paper to relate Poisson-bracket estimates to multidegrees of tame automorphisms. The source's abstract states that the authors give a counterexample to this conjecture, so it is refuted.