Stoll's very good curves conjecture

Let $k$ be a number field, and let $C$ be a smooth projective geometrically connected curve over $k$. In the paper, a curve is very good when its finite descent obstructions account precisely for the rational points, in the sense defined earlier. Stoll's very good curves conjecture. Every such curve $C$ is very good. The conjecture is motivated by the many known examples of higher-genus curves that are excellent with respect to abelian coverings, including positive-genus modular curves. Its general validity is left open.