Kim's conjecture on eventual equality of nonabelian Chabauty sets

Let $S$) be a finite set of primes, let $X/\mathbb{Z}_S$ be a smooth hyperbolic curve, and let $p$ be a prime not in $S$. For each positive integer $n$, let $X(\mathbb{Z}_p)_n$ denote the depth-$n$ nonabelian Chabauty set, with $X(\mathbb{Z}_p)_n=X(\mathbb{Z}_p)_{\emptyset,n}$.

Kim's conjecture. For $n\gg 0$,

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

This conjecture predicts that sufficiently deep nonabelian Chabauty sets recover the integral points exactly. Kim's programme gives finiteness of these sets under suitable dimension comparisons, while the asserted equality for all sufficiently large depths remains open in the stated context.