The generalized Mazur conjecture for rings of localized rationals

Let $P$ be the set of all primes and let $W_Q e P$ be a \subset of $P$. Define

$$O_{Q,W_Q}:=\{x/y:x\in\mathbb{Z},\ y\in\langle W_Q\rangle\}.$$

For a variety $V$ over $\mathbb{Q}$, write $V(O_{Q,W_Q})$ for its points over this ring. Generalized Mazur conjecture. The set $V(O_{Q,W_Q})$ has finitely many connected components. This extends the rational-points formulation to rings obtained by allowing denominators generated by a set of primes; the source gives no resolution status for this formulation.