The $p$-adic section conjecture for curves

Let $k/\mathbb Q_p$ be a finite extension, and let $X/k$ be a smooth, projective and geometrically connected curve of genus at least $2$. A rational point $a\in X(k)$ determines a section $s_a:\operatorname{Gal}_k\to\pi_1(X)$, and sections are considered up to conjugacy by $\pi_1(\overline{X})$. The $p$-adic section conjecture. The map

is a bijection from $X(k)$ onto the set of $\pi_1(\overline{X})$-conjugacy classes of sections of $\pi_1(X/k)$. This is the local analogue of Grothendieck's section conjecture. The paper describes evidence for it, but the asserted bijectivity remains open.