The trivial-centralizer conjecture for simple Shamsuddin derivations

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

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

is called a Shamsuddin derivation. A derivation is simple if the only stable ideals of $K[x]$ are $0$ and $K[x]$. Define

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

The trivial-centralizer conjecture. If $D$ is a simple Shamsuddin derivation of $K[x]$, then

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

The source notes that the assertion is known for $n=2$, while the general case considered there remains a conjecture.