Grothendieck's section conjecture for hyperbolic curves

Let $V/k$ be a smooth, geometrically connected, hyperbolic curve over a number field $k$, with smooth completion $V\subset X$ and boundary $Y=X\setminus V$. For each $k$-rational cusp $y\in Y(k)$, let $V_y=V\times_X\operatorname{Spec}(\mathcal{O}_{X,y}^h)$ be the scheme of nearby points, and let $\mathscr{S}_{\pi_1(V/k)}$ denote the set of sections of $1\to\pi_1(\overline{V})\to\pi_1(V)\to\operatorname{Gal}_k\to1$, considered up to $\pi_1(\overline{V})$-conjugacy. The non-abelian Kummer map is denoted by $\kappa$. Grothendieck's section conjecture. The map

$$\kappa: V(k)\sqcup\bigsqcup_{y\in Y(k)}\mathscr{S}_{\pi_1(V_y/k)}\longrightarrow\mathscr{S}_{\pi_1(V/k)}$$

is bijective. Thus every section is either explained by a unique rational point or belongs to a packet of cuspidal sections based at a unique rational cusp. The conjecture is widely open in this generality: injectivity is known, while surjectivity remains the difficult part. The source notes that the conjecture predicts, in particular, that every Selmer section is rational or cuspidal.