A field $K$ is called ample if every smooth curve over $K$ that has a $K$-point has infinitely many $K$-points. Examples include fraction fields of henselian rings.

I have known for a long time that $\mathbb{Q}^{ab}$ is conjectured to be ample, but I don't know how much evidence there is for this conjecture. Can you direct me to some references supporting this conjecture, or suggest heuristic arguments that would explain why this conjecture is reasonable?