Harari–Voloch conjecture for integral points on rational hyperbolic curves

Let $F$ be a global field, let $X_F$ be a hyperbolic rational curve, meaning an open subset of $\mathbf P^1_F$ whose complement is a reduced separable divisor of degree at least $3$, and let $\mathbf X$ be an integral model of $X_F$ over $\mathfrak{o}_S$, where $S$ is a non-empty finite subset of $\Omega_F$ containing $\infty_F$. Define

$$B_S(X_F):=\operatorname{Ker}\left[\operatorname{Br}(X_F)\rightarrow\prod_{v\in S}\operatorname{Br}(X_{F_v})/\operatorname{Br}(F_v)\right].$$

Harari–Voloch conjecture. The diagonal map

$$\mathbf X(\mathfrak{o}_S)\longrightarrow\left(\prod_{v\notin S}\mathbf X(\mathfrak{o}_v)\right)^{B_S(X_F)}$$

is bijective. Harari and Voloch proposed this as an integral-point analogue of the Brauer–Manin principle for rational hyperbolic curves. The paper proves the conjecture over global function fields, while the general formulation is not asserted here to be resolved.