Infinite-point conjecture for pointed nonsingular curves over finitely generated fields

Let $K$ be a field finitely generated over $\mathbb{Q}$, let $X$ be a pointed nonsingular curve over $K$, and let $\bar K(\boldsymbol{\sigma})$ denote a field obtained by fixing a tuple $\boldsymbol{\sigma}$ in an algebraic closure. The infinite-point conjecture. For every tuple $\boldsymbol{\sigma}$, the curve $X$ has infinitely many $\bar K(\boldsymbol{\sigma})$-points. This is presented as a consequence of the conjecture that fields with finitely generated Galois group are ample, and remains open in the stated generality.