Are there native counterexamples in higher dimensions?

Padding makes every three-dimensional counterexample a four-dimensional one. The question is whether dimension four and up has anything new.

Two candidates are open.

• Harish's map of ℂ⁴ has fiber degree 8, which rules out a padded Alpöge map, of degree 3. It does not rule out a padded three-dimensional map of fiber degree 8, and Gallagher's $F_8$ is one.
• Gao's $F_4$ has degree 5. It is proved inequivalent to $\Phi\times\mathrm{id}$ only for $\Phi$ of a different degree, and he leaves the degree-5 case open.^[1]

Geometric degree is invariant under the equivalence, so it cannot settle the question alone. A proof would need a finer invariant, such as the non-properness set or the grading, compared across source and target. This question is left open by Gao's tangent-sweep paper for its own four-dimensional maps.

1. ↑ S. Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222, Theorem 4.3 and the remark after it.