Is $mathrm{End}-{0}=mathrm{Aut}$ for derivation Lie algebra?

Is it true that every nonzero endomorphism of Lie $mathbb{C}$-algebra $mathbb{C}[x_1,ldots, x_n]partial_{x_1}oplusldotsoplusmathbb{C}[x_1,ldots, x_n]partial_{x_n}$ is an automorphism?

As I know a positive answer to question implies the Jacobian conjecture for $mathbb{A}_n$.

So is it equivalent or a stronger result than JC?

Idea of proof that (positive answer to Question) $Rightarrow$ JC

If $(F_1,ldots, F_n)$ is a set of polynomials with constant nonzero Jacobian, then $mathbb{C}[F_1,ldots, F_n]subseteqmathbb{C}[x_1,ldots, x_n]$ is subalgebra. It can be shown that the constant Jacobian means that each derivation from $mathbb{C}[F_1,ldots, F_n]$ can be uniquely continued to a derivation of $mathbb{C}[x_1,ldots, x_n]$. So we have the inclusion $text{Der}(mathbb{C}[F_1,ldots, F_n])totext{Der}(mathbb{C}[x_1,ldots, x_n])$. If the main question has a positive answer, then this inclusion is surjective and there are derivations $D_1,ldots, D_n$ of $mathbb{C}[x_1,ldots, x_n]$ with $D_i|_{mathbb{C}[F_1,ldots, F_n]}=partial_{F_i}$. We know that $F_i$ are slices of $D_i$, so it is enough to check that $D_i$ are locally nilpotent commuting derivations. Since $D_i$ are locally nilpotent on $mathbb{C}[F_1,ldots, F_n]$ one can prove that $text{ad}D_i|_{text{Der}(mathbb{C}[F_1,ldots, F_n])}$ are locally nilpotent derivations of $text{Der}(mathbb{C}[F_1,ldots, F_n])$, so (from Question statement) $text{ad}D_i$ is locally nilpotent on $text{Der}(mathbb{C}[x_1,ldots,x_n])$. Thus one can prove that $D_i$ is locally nilpotent on $mathbb{C}[x_1,ldots, x_n]$ and the $D_i$ are locally nilpotent with slices $F_i$ which commute with each other, so we have JC.