A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves

We state a conjectural criterion for identifying global integral points on a hyperbolic curve over $\mathbb{Z}$ in terms of Selmer schemes inside non-abelian cohomology functors with coefficients in ${\mathbb{Q}}_p$-unipotent fundamental groups. For ${\mathbb{P}}^1\setminus \{0,1,\mathrm{\infty }\}$ and the complement of the origin in semi-stable elliptic curves of rank 0, we compute the local image of global Selmer schemes, which then allows us to numerically confirm our conjecture in a wide range of cases.