Kim's finite-level cohomological globalization conjecture for integral points

Let $\mathcal{X}$ be a hyperbolic curve over the relevant arithmetic base, let $p$ be a prime, and let $\mathcal{X}(\mathbb{Z})\subset\mathcal{X}(\mathbb{Z}_p)$ denote the global and $p$-adic integral points. For each level $n$, define the set of points that are cohomologically global of level $n$ by

$$\mathcal{X}(\mathbb{Z}_p)_n:=j_p^{-1}(\operatorname{loc}_p[H^1_{\mathbb{Z}}(G,U_n)]).$$

Here $U_n$ is the $n$th quotient of the unipotent fundamental group, $H^1_{\mathbb{Z}}(G,U_n)$ is the Selmer scheme, and $j_p$ and $\operatorname{loc}_p$ are the local Albanese and localization maps. Kim's conjecture. For sufficiently large $n$,

$$\mathcal{X}(\mathbb{Z}_p)_n=\mathcal{X}(\mathbb{Z}).$$

The conjecture says that sufficiently deep non-abelian cohomological conditions recognize exactly the global integral points. Since the sets $\mathcal{X}(\mathbb{Z}_p)_n$ should be computable in principle, it suggests a method for computing $\mathcal{X}(\mathbb{Z})$.