Is it true that every not zero endomorphism of Lie $\mathbb{C}$-algebra $\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\partial_{x_n}$ is an automorphism?

As I know this question implies the Jacobian conjecture for $\mathbb{A}_n$.

So is it equivalent or a stronger result than JC?