Stoll's conjecture on weak approximation with Brauer–Manin obstruction for curves

Let $K$ be a number field, let $\infty_K$ denote the set of archimedean places of $K$, and let $C$ be a smooth, projective, geometrically connected curve over $K$. Write $C(\mathbb{A}_K)^{\operatorname{Br}}$ for the Brauer–Manin set and $\operatorname{pr}^{\infty_K}$ for the projection from adelic points to the adeles with the archimedean components removed.

Stoll's conjecture. The set $C(K)$ is dense in

$$\operatorname{pr}^{\infty_K}\bigl(C(\mathbb{A}_K)^{\operatorname{Br}}\bigr).$$

In particular, if $C(K)$ is finite, then

$$C(K)=\operatorname{pr}^{\infty_K}\bigl(C(\mathbb{A}_K)^{\operatorname{Br}}\bigr).$$

This conjecture predicts weak approximation with Brauer–Manin obstruction off the archimedean places for every smooth, projective, geometrically connected curve over a number field. The source attributes it to Stoll and does not state that it has been resolved.