The Second Kernel Conjecture

Let $K$ be a field of characteristic zero and let $K[T]=K[t_1,\dots,t_n]$. A derivation $D$ of a ring is locally nilpotent if for every $a$ there is an integer $m(a)$ such that $D^{m(a)}a=0$. Second Kernel Conjecture. If $D$ is a locally nilpotent derivation on $K[T]$ and there is a polynomial $p\in K[T]$ with $Dp=1$, then

$$\ker(D)=K[F_1,\dots,F_{n-1}]$$

for some $F_i\in K[T]$ algebraically independent over $K$. This is another kernel problem related in the source to the Jacobian and Dixmier questions; its resolution is not given.