The toric section conjecture for hyperbolic curves over $p$-adic fields

Let $k$ be a finite extension of $\mathbb{Q}_{p}$, and let $X$ be a hyperbolic curve over $k$, meaning a smooth projective geometrically connected curve of genus at least $2$. Its toric Kummer map is

$$X(k)\to\mathscr{S}^{\mathrm{tor}}_{X/k}.$$

Toric $p$-adic section conjecture. This map is bijective. The statement is supported by the paper's proof of the corresponding result for torsors under abelian varieties over finite extensions of $\mathbb{Q}_{p}$, but remains conjectural for hyperbolic curves in general.