  Let $K$ be a number field not containing a CM subfield. For any smooth
projective curve $Y/K$ of genus $\geq2$, we prove that the image of the
"Selmer" part of Grothendieck's section set inside the $K_v$-rational points
$Y(K_v)$ is finite for every finite place $v$. This gives an unconditional
verification of a prediction of Grothendieck's section conjecture. In the
process of proving our main result, we also refine and extend the method of
Lawrence and Venkatesh, with potential consequences for explicit computations.
