The translation-centralizer conjecture for simple polynomial derivations

Let $K$ be a field of characteristic zero and let $K[x]=K[x_1,\ldots,x_n]$. Consider the derivation

$$D=\sum_{i=1}^n p_i(x)\partial_{x_i},\qquad p_i(x)\in K[x].$$

Let $\operatorname{Aut}(K[x])_D$ denote the subgroup of polynomial automorphisms commuting with $D$, and let $T$ be the translation subgroup of $\operatorname{Aut}(K[x])$. The translation-centralizer conjecture. If $D$ is simple, then

$$\operatorname{Aut}(K[x])_D=T.$$

The source proposes this as a conjecture for all simple derivations. Earlier examples in the paper show that the analogous finiteness claim fails in dimensions at least three, while the conjecture here allows precisely the translation symmetries.