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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?

Idea of proof that $Question=> JC$

If $(F_1,\ldots, F_n)$ is set of polynomials with constant not zero Jacobian then $\mathbb{C}[F_1,\ldots, F_n]\subseteq\mathbb{C}[x_1,\ldots, x_n]$ 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 the derivation of $\mathbb{C}[x_1,\ldots, x_n]$. So we have the inclusion $\text{Der}(\mathbb{C}[F_1,\ldots, F_n])\to\text{Der}(\mathbb{C}[x_1,\ldots, x_n])$. If the main question is correct 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 we have $D_i$ are locally niplotent with slices $F_i$ which commute with each other so we have JC.

Is it true that every nonzero 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 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 $\mathbb{C}[F_1,\ldots, F_n]\subseteq\mathbb{C}[x_1,\ldots, x_n]$ 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 $\text{Der}(\mathbb{C}[F_1,\ldots, F_n])\to\text{Der}(\mathbb{C}[x_1,\ldots, x_n])$. 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.

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?

Idea of proof that $Question=> JC$

If $(F_1,\ldots, F_n)$ is set of polynomials with constant not zero Jacobian then $\mathbb{C}[F_1,\ldots, F_n]\subseteq\mathbb{C}[x_1,\ldots, x_n]$ is subalgebra. It can be shown that the constant Jacobian means that each derivation from $\mathbb{C}[F_1,\ldots, F_n]$ can be continued to the derivation of $\mathbb{C}[x_1,\ldots, x_n]$. So we have the inclusion $\text{Der}(\mathbb{C}[F_1,\ldots, F_n])\to\text{Der}(\mathbb{C}[x_1,\ldots, x_n])$. If the main question is corrext 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 derivations. Since $D_i$ are locally nilpotent on $\mathbb{C}[F_1,\ldots, F_n]$ one can prove that $\text{ad}D_i|_{\mathbb{C}[F_1,\ldots, F_n]}$ are locally nilpotent derivations of $\text{Der}(\mathbb{C}[F_1,\ldots, F_n])$ so $\text{ad}D_i$ 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 we have $D_i$ are locally niplotent with slices $F_i$ which commute with each other so we have JC.

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?

Idea of proof that $Question=> JC$

If $(F_1,\ldots, F_n)$ is set of polynomials with constant not zero Jacobian then $\mathbb{C}[F_1,\ldots, F_n]\subseteq\mathbb{C}[x_1,\ldots, x_n]$ 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 the derivation of $\mathbb{C}[x_1,\ldots, x_n]$. So we have the inclusion $\text{Der}(\mathbb{C}[F_1,\ldots, F_n])\to\text{Der}(\mathbb{C}[x_1,\ldots, x_n])$. If the main question is correct 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 we have $D_i$ are locally niplotent with slices $F_i$ which commute with each other so we have JC.

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?

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?

Idea of proof that $Question=> JC$

If $(F_1,\ldots, F_n)$ is set of polynomials with constant not zero Jacobian then $\mathbb{C}[F_1,\ldots, F_n]\subseteq\mathbb{C}[x_1,\ldots, x_n]$ is subalgebra. It can be shown that the constant Jacobian means that each derivation from $\mathbb{C}[F_1,\ldots, F_n]$ can be continued to the derivation of $\mathbb{C}[x_1,\ldots, x_n]$. So we have the inclusion $\text{Der}(\mathbb{C}[F_1,\ldots, F_n])\to\text{Der}(\mathbb{C}[x_1,\ldots, x_n])$. If the main question is corrext 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 derivations. Since $D_i$ are locally nilpotent on $\mathbb{C}[F_1,\ldots, F_n]$ one can prove that $\text{ad}D_i|_{\mathbb{C}[F_1,\ldots, F_n]}$ are locally nilpotent derivations of $\text{Der}(\mathbb{C}[F_1,\ldots, F_n])$ so $\text{ad}D_i$ 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 we have $D_i$ are locally niplotent with slices $F_i$ which commute with each other so we have JC.