The kernel conjecture for Jacobian derivations

Let $K$ be a field of characteristic zero and let $F=(F_1,\dots,F_n):K^n\to K^n$ be a polynomial function with $\det(JF)\in K^*$. Define derivations on $K[T]$ by

$$\begin{bmatrix}\frac{\partial}{\partial F_1}\\vdots\\frac{\partial}{\partial F_n}\end{bmatrix}=((JF)^{-1})^T\begin{bmatrix}\frac{\partial}{\partial t_1}\\vdots\\frac{\partial}{\partial t_n}\end{bmatrix}.$$

Kernel conjecture.

$$\ker\left(\frac{\partial}{\partial F_n}\right)=K[F_1,\dots,F_{n-1}].$$

This conjecture is presented as related to the Jacobian and Dixmier conjectures; the supplied text gives no resolution.