Convergence of polylogarithmic Chabauty–Kim loci

Let $Z$ be a totally real integer scheme, and let $\mathfrak{p} in Z$ be a totally split prime. For each $n$, let $X(\mathcal{O}_\mathfrak{p})_n$ denote the associated polylogarithmic Chabauty–Kim locus. Convergence conjecture. For $n$ sufficiently large,

$$X(\mathcal{O}_\mathfrak{p})_n=X(Z).$$

This predicts that the nested polylogarithmic Chabauty–Kim loci eventually recover exactly the integral points. The preceding results establish finiteness for sufficiently large $n$ under the same totally real and totally split hypotheses, but not equality with $X(Z)$.