Junker–Koenigsmann conjecture on ample fields

A field $K$ is ample if every nonsingular curve over $K$ has either no $K$-points or infinitely many. Junker–Koenigsmann's conjecture. Every infinite field with finitely generated absolute Galois group is ample. This would explain why fields arising from finitely generated Galois groups should have abundant rational points, although the conjecture remains open.