$\Sigma$-refined Kim's conjecture for the thrice-punctured line

Let $S$ be a finite set of primes, let $X=\mathbb P^1\smallsetminus\{0,1,\infty\}$, and let $p\notin S$ be a prime. A refinement condition is a tuple $\Sigma=(\Sigma_\ell)_{\ell\in S}\in\{0,1,\infty\}^S$, and let $X(\mathbb Z_S)_\Sigma$ be the set of $S$-integral points whose reduction modulo each $\ell\in S$ lies in $(X\cup\{\Sigma_\ell\})(\mathbb F_\ell)$. Let $X(\mathbb Z_p)_{S,n}^{\Sigma}$ be the associated $\Sigma$-refined Chabauty--Kim locus. $\Sigma$-refined Kim's conjecture. One has

$$X(\mathbb Z_p)_{S,n}^{\Sigma}=X(\mathbb Z_S)_\Sigma$$

for $n\gg0$. The total refined locus is the union of these loci over all refinement conditions, so this conjecture refines the assertion that deep Chabauty--Kim loci contain no extra $p$-adic points. Its general status is not resolved by the supplied text.