Podewski's conjecture on infinite model-theoretically minimal fields

Let $K$ be an infinite field. It is model-theoretically minimal if every definable subset of $K$ is finite or cofinite. Podewski's conjecture. If $K$ is model-theoretically minimal, then $K$ is algebraically closed.

Podewski's conjecture connects model-theoretic minimality with algebraic closure. It was proven in positive characteristic by Wagner; the characteristic-zero case remains open. A counterexample would also imply that the finite-closed topology on a perfect field is discrete.