Bertoncello–Levcovitz conjecture on Shamsuddin derivations and isotropy groups

Let $K[x_1,x_2,\ldots,x_n]$ be a polynomial ring, and let $D$ be a Shamsuddin derivation, meaning a derivation of the form

$$D=\partial_x+\sum_{i=1}^s\sum_{j=1}^{r_i}(a_i y_{i,j}+b_{i,j})\partial_{i,j},$$

where $a_i,b_{i,j}\in K[x]$ and $a_i\neq a_l$ for $i\neq l$. Write

$$\operatorname{Aut}(K[x_1,x_2,\ldots,x_n])_D=\{\rho\in\operatorname{Aut}(K[x_1,x_2,\ldots,x_n])\mid \rho D=D\rho\}$$

for the isotropy subgroup of $D$. Bertoncello–Levcovitz conjecture. If $D$ is a Shamsuddin derivation of $K[x_1,x_2,\ldots,x_n]$, then $D$ is simple if and only if

$$\operatorname{Aut}(K[x_1,x_2,\ldots,x_n])_D=\{\operatorname{id}\}.$$

This conjecture proposes that simplicity of a Shamsuddin derivation is equivalent to triviality of its isotropy group. The source notes that triviality of the isotropy group had already been proved for simple Shamsuddin derivations, while the converse remains the proposed direction.