Let $X$ be a smooth geometrically integral projective variety over $\mathbb{Q}$. Then we may consider the closure $\overline{X(\mathbb{Q})}$ of $X(\mathbb{Q})$ inside the adelic points $X(\mathbb{A})=\prod_v X(\mathbb{Q}_v)$ of $X$. However, we may also take the closure $\overline{X(\mathbb{Q})}^v$ of $X(\mathbb{Q})$ inside $X(\mathbb{Q}_v)$ for any place $v$ of $\mathbb{Q}$. Obviously we have $$\overline{X(\mathbb{Q})} \subset \prod_v \overline{X(\mathbb{Q})}^v \subset X(\mathbb{A}).$$

My question whether this first inequality is actually an equality?

My motivation is that I am trying to understand better $\overline{X(\mathbb{Q})}$ and what it looks like. I will simply note that the answer to my question is yes in the easy cases where $X$ satisfies weak approximation and when $X(\mathbb{Q})$ is empty.

Edit: To make sure there are not simple counter-examples like the one David pointed out below, I am assuming that $X(\mathbb{Q})$ is Zariski dense. I should also note that I am particularly interested in the case where $X$ is a fano variety.

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