The Jacobian Conjecture over algebraically closed fields of characteristic zero

Let $n$ be a positive integer, let $k$ be an algebraically closed field of characteristic zero, and let $f:k^n\rightarrow k^n$ be a polynomial map, meaning that each component $f_i:k^n\rightarrow k$ is polynomial. Let $J(f)$ denote the Jacobian of $f$. The Jacobian Conjecture. If $J(f)=1$, then $f$ is invertible with polynomial inverse. The source notes that the complex case suffices by the Lefschetz Principle, and that the conjecture remains open apart from the trivial one-dimensional case.