As was pointed out by Felipe Voloch in the comments, this obstruction is equivalent to the finite abelian descent obstruction [Stoll, **Thm. 6.4** and the discussion before **Cor. 6.2**]. It is also shown that for curves, this agrees with the Brauer–Manin obstruction [Stoll, **Cor. 7.3**]. Finally, Stoll conjectures the following:

**Conjecture** [Stoll, **Conj. 9.1**]. *Let $C$ be a smooth projective geometrically connected curve over a number field $K$. Then $\overline{C(K)} = C(\mathbf A_K)_\bullet^{\text{f-ab}}$ (see [Stoll, **Def. 5.4** and §2] for notation).*

That is, the finite abelian descent obstruction is the only obstruction to the Hasse principle and weak approximation.

**What is known.**

 - If $g = 0$, then $C$ satisfies the Hasse principle $C(K) = \varnothing \Leftrightarrow C(\mathbf A_K)_\bullet = \varnothing$. If $C(\mathbf A_K)_\bullet = \varnothing$ then there is nothing to prove, and if $C \cong \mathbf P^1_K$ then $C(K)$ is dense in $C(\mathbf A_K)_\bullet$.
 - If $g = 1$, then the conjecture is true if and only if $Ш(K,\operatorname{Pic}^0_C)_{\text{div}} = 0$ [Stoll, **Cor. 6.2** and the discussion after **Cor. 6.3**]. In particular, this is a weak form of the Tate–Shafarevich conjecture.
 - If $g \geq 2$, then $C(K)$ is finite by Faltings, so $\overline{C(K)} = C(K)$. Thus, the conjecture says that $C(\mathbf A_K)_\bullet^{\text{f-ab}} = C(K)$. I'm not even sure if we know whether $C(\mathbf A_K)_\bullet^{\text{f-ab}}$ is finite.

Finally, Stoll proves the conjecture in the following case:

**Theorem** [Stoll, **Thm. 8.6**]. *Assume $C$ admits a nonconstant map $C \to A$ into an abelian variety $A$ such that $A(K)$ is finite and $Ш(K,A)_{\text{div}} = 0$. Then $C(K) = C(\mathbf A_K)_\bullet^{\text{f-ab}}$.*

This is for example the case when $K = \mathbf Q$ and $A$ is modular of analytic rank $0$. As mentioned before, the condition $Ш(K,A)_{\text{div}}$ is expected to always be true, but the condition that $A(K)$ is finite is a serious restriction on the generality of the theorem.

**Recent progress?**

I haven't worked on rational points for a while, so I don't know what the current status is. I would be surprised if there were a great breakthrough on the conjecture in its general form, though.

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**References.**

[Stoll] <cite authors="Stoll, Michael">Stoll, Michael, [*Finite descent obstructions and rational points on curves*](http://dx.doi.org/10.2140/ant.2007.1.349), Algebra Number Theory 1, No. 4, 349-391 (2007). [ZBL1167.11024](https://zbmath.org/?q=an:1167.11024).</cite>