The converse conjecture for simplicity and trivial centralizers of Shamsuddin derivations

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

$$D=\partial_{x_1}+\sum_{i=2}^n(a_i x_i+b_i)\partial_{x_i},\qquad a_i,b_i\in K[x_1],$$

be a Shamsuddin derivation. Define

$$\operatorname{Aut}(K[x])_D=\{\rho\in\operatorname{Aut}(K[x])\mid \rho D=D\rho\}.$$

The converse conjecture. The derivation $D$ is simple if and only if

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

The source presents this as a further conjecture after noting results establishing the trivial-centralizer direction for simple Shamsuddin derivations. Its general status is not resolved in the supplied text.