Adjamagbo's separable Jacobian conjecture

Let $k$ be a field of characteristic $p>0$, let $X=(x_1,\dots,x_n)$, and let $F=(F_1,\dots,F_n)in k[X]^n$. Write

$$k(F)=k(F_1,\dots,F_n)\subseteq k(X)=k(x_1,\dots,x_n).$$

Adjamagbo's separable Jacobian conjecture. Assume that

$$\det \operatorname{Jac}(F)\in k^\times \qquad\text{and}\qquad p\nmid [k(X):k(F)].$$

Then $F$ is a polynomial automorphism of $\mathbb{A}_k^n$.

This positive-characteristic refinement of the Jacobian conjecture was proposed by Adjamagbo and is intended to capture the separable case. The stated conjecture is refuted by the paper's explicit characteristic-$2$ example in dimension $3$, whose field extension has degree $3$ but whose polynomial map is noninjective; stabilization gives counterexamples in every dimension $n\geq 3$.