The finiteness conjecture for centralizers of simple derivations in two variables

Let $K$ be a field of characteristic zero and let $K[x_1,x_2]$ be the polynomial algebra in two variables. A $K$-derivation $D$ is simple if the only $D$-stable ideals of $K[x_1,x_2]$ are $0$ and $K[x_1,x_2]$. Define

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

The finiteness conjecture. If $D$ is a simple derivation of $K[x_1,x_2]$, then

$$\operatorname{Aut}(K[x_1,x_2])_D$$

is finite.

The conjecture was proposed by the authors cited in the source. Its status is not resolved in the supplied text.