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. For each depth $n\geq 1$, write $X(\mathbb Z_p)_{S,n}^{\min}$ for the refined Chabauty--Kim locus and let $X(\mathbb Z_S)$ denote the set of $S$-integral points. Refined Kim's conjecture. One has

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

for $n\gg0$. The refined locus contains the $S$-integral points by construction; the conjecture asserts that sufficiently deep refined Chabauty--Kim computations recover exactly the integral points. The paper verifies the corresponding conjecture over $\mathbb Z[1/6]$ computationally for all auxiliary primes $p<10{,}000$, but the general assertion remains open.