Finiteness conjecture for automorphism groups of derivations with zero eigenvalue degree

Let $D$ be a nonzero derivation of ${\rm K}[x,y]$, where ${\rm K}$ is the underlying field, and let ${\mathbb e}(D)$ denote the eigenvalue-degree invariant used in the paper. Finiteness conjecture. If

$${\mathbb e}(D)=0,$$

then ${\rm Aut}(D)$ is finite. The paper notes that this is true when ${\mathbb K}={\mathbb C}$, but its general validity is presented as a conjecture.