Poonen's finite-place obstruction conjecture for curves

Let $C$ be a smooth projective geometrically connected curve of genus at least $2$ over a number field $k$, with $C(k)=\emptyset$. Assume that $C$ has a rational divisor class of degree $1$, and let $J$ be its Jacobian; the divisor class induces an embedding

$$\iota:C\to J.$$

For a finite place $v$ of good reduction, write $\mathbb F_v$ for the residue field. Poonen's finite-place obstruction conjecture. There is a finite set $S$ of finite places of good reduction for $C$ such that

$$\operatorname{im}\left(J(k)\to\prod_{v\in S}J(\mathbb F_v)\right)\cap\prod_{v\in S}\iota\left(C(\mathbb F_v)\right)=\emptyset.$$

This stronger conjectural form is motivated by heuristic arguments of Poonen and would give a finite collection of reductions witnessing the absence of a rational point. The paper reports computational evidence for genus-$2$ curves over $\mathbb Q$, but the assertion remains open in general.