Finiteness conjecture for birational symmetries of simple foliations

Let ${\mathbb K}={\mathbb C}$, and let $D$ be a derivation of ${\mathbb C}[x,y]$. Assume that $D$ is simple, meaning that it does not stabilize a nontrivial ideal. Let ${\mathcal F}_D$ be the algebraic foliation on ${\mathbb C}^2$ defined by $D$, and let ${\rm Bir}({\mathcal F}_D)$ be the group of birational maps of ${\mathbb C}^2$ that stabilize ${\mathcal F}_D$.

Finiteness conjecture. If $D$ is simple, then

$${\rm Bir}({\mathcal F}_D)\text{ is finite}.$$

This would constrain the birational symmetry groups of algebraic foliations arising from simple derivations. The paper states the claim as a conjecture to be considered in forthcoming work, and no resolution is supplied here.