Finite descent conjecture for curves

Let $C$ be a curve over ${\mathbb Q}$, and let $\operatorname{Sel}^{(n)}(C)$ denote the $n$-Selmer set, consisting of the $n$-coverings of $C$ with points everywhere locally. Finite descent conjecture. If

$$C({\mathbb Q})=\emptyset,$$

then

$$\operatorname{Sel}^{(n)}(C)=\emptyset$$

for some $n\geq 1$. This would imply that sufficiently high descent can certify the nonexistence of rational points, extending the effectiveness observed for many small genus-two curves; the source gives reasons for the conjecture but no resolution.