The toric section conjecture for hyperbolic curves over finitely generated fields

Let $k$ be a finitely generated extension of $\mathbb{Q}$, 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 section conjecture. This map is bijective. The conjecture is proposed as an intermediate step toward Grothendieck's section conjecture; the paper proves injectivity of the natural map from toric to étale sections in relevant cases and proves the analogous toric statement for torsors under abelian varieties over $p$-adic fields.