The local-global equality conjecture for varieties over global function fields

Let $K$ be a global function field over a finite field $k$ of positive characteristic, let $S$ be a finite set of places of $K$, and let $\Gamma\subset O_S^*$ be a subgroup. Fix a cofinite subset $\Sigma\subset\Sigma_K$, and let $W$ be a closed $K$-variety in $\mathbb A^M$. Writing $W(\overline{\Gamma})$ for the points of $W$ whose coordinates lie in the product of the local closures of $\Gamma$, and $\overline{W(\Gamma)}$ for the topological closure of the diagonal image of $W(\Gamma)$, local-global equality conjecture.

$$W(\overline{\Gamma})=\overline{W(\Gamma)}.$$

This conjecture asks whether local points obtained by independently taking limits of $\Gamma$-coordinates are exactly the limits of global $\Gamma$-points on every closed variety over $K$. The supplied text does not state whether the equality is known or remains open.