The finite-characteristic reformulation of the Jacobian Conjecture

Let $n\ge 2$, let $d\ge 1$, and let ${\bf k}$ be a field. A polynomial map $\phi:{\mathbb A}^n_{\bf k}\to{\mathbb A}^n_{\bf k}$ has degree at most $d$ if its coordinate polynomials do. Finite-characteristic Jacobian Conjecture. For every $d\ge 1$, there exist $d'\ge 1$ and a finite set of primes $S$ such that, for every field ${\bf k}$ with

$$\operatorname{char}{\bf k}\notin S\cup\{0\},$$

and every polynomial map $\phi:{\mathbb A}^n_{\bf k}\to{\mathbb A}^n_{\bf k}$ of degree at most $d$ with Jacobian $1$, the inverse map exists and has degree at most $d'$. The paper states that this finite-characteristic formulation is equivalent to $\operatorname{JC}_n$; the displayed condition is therefore unresolved precisely when the Jacobian Conjecture is unresolved.