Mazur's topology of rational points conjecture

Let $V$ be any variety over $\mathbb{Q}$. The topological closure of $V(\mathbb{Q})$ in $V(\mathbb{R})$ is required to have at most finitely many connected components.

Mazur's conjecture. The topological closure of $V(\mathbb{Q})$ in $V(\mathbb{R})$ possesses at most a finite number of connected components.

This conjecture concerns the archimedean topology of rational points and has consequences for Diophantine definitions over $\mathbb{Q}$, including the nonexistence of Diophantine definitions for infinite discrete sets such as $\mathbb{Z}$. Its resolution status is not specified in the source.