Ampleness of the maximal abelian extension of the rationals

Definitions and mathematical statement

Inside a fixed algebraic closure of $\mathbb Q$, let $\mathbb Q_{\rm ab}$ be the union of the finite Galois extensions of $\mathbb Q$ whose Galois groups are abelian.

A field $K$ is called ample if every smooth geometrically integral curve over $K$ that has a $K$-rational point has infinitely many such points. Here a curve is a separated scheme of finite type and dimension one; geometric integrality means integrality after extension to an algebraic closure of $K$.

Is $\mathbb Q_{\rm ab}$ ample? That is, for every such curve $C$ over this field, does \[ C(\mathbb Q_{\rm ab})\ne\varnothing \quad\Longrightarrow\quad \#C(\mathbb Q_{\rm ab})=\infty \] hold?

Short English statement

Over the maximal abelian extension of the rationals, does one rational point on a smooth geometrically integral curve guarantee infinitely many?

Sources

L. Bary-Soroker and A. Fehm, Open Problems in the Theory of Ample Fields (2011), Question I and Proposition 2.1, https://arxiv.org/abs/1106.1310. For later context see E. Walsberg, Topological proofs of results on large fields, Comptes Rendus Mathématique 360 (2022), 1187--1192, introduction, https://doi.org/10.5802/crmath.305. This is an independently written formulation of the mathematical question.