Potential density of integral points in relative character varieties

Let $Y$ be a smooth complex variety equipped with a smooth projective simple normal crossings compactification $\overline{Y}$, and let $D=\overline{Y}\setminus Y$ have components $D_1,\ldots,D_n$. Let $G$ be a Chevalley group over $\mathbb{Z}$, let $K$ be a number field with ring of integers $\mathscr{O}_K$, and let $X_{G,\underline{C}}(Y)$ denote the relative character variety of representations of $\pi_1(Y)$ into $G$ whose local monodromy around each $D_i$ has prescribed image $C_i$ in $(G/_{\operatorname{ad}}G)(\mathscr{O}_K)$, where $\underline{C}=(C_i)_{i=1,\ldots,n}$. Fix

$$\underline{C}\in (G/_{\operatorname{ad}}G)(\mathscr{O}_K)^n.$$

Potential-density conjecture. Integral points are potentially Zariski-dense in the $K$-scheme $X_{G,\underline{C}}(Y)_K$: there exists a finite extension $L/K$ such that the Zariski closure of the $\mathscr{O}_L$-points of $X_{G,\underline{C}}(Y)$ contains $X_{G,\underline{C}}(Y)_K$.

The conjecture predicts arithmetic largeness of relative character varieties with fixed boundary monodromy, generalizing the paper's evidence from the case of $\operatorname{SL}_2$ and using a reduction to Riemann surfaces together with mapping-class-group dynamics. Its general validity for Chevalley groups and arbitrary smooth quasi-projective varieties remains open.