Let $V$ be a (sufficiently nice) projective variety of dimension $n$ over $\mathbb{Q}$, and let $V_1,\ldots,V_k$ be finitely many subvarieties of $V$ of strictly lower dimensions. Let $K/\mathbb{Q}$ be a finitely generated field extension. Recall that a place of $K/\mathbb{Q}$ is a field of characteristic 0 that arises as the quotient field of a valuation ring of $K$.
Assume that the variety $V$ has a $K$-rational point, but the subvarieties $V_1, \ldots, V_k$ do not. Is there always a non-trivial place $E/\mathbb{Q}$ of $K/\mathbb{Q}$ such that $V$ has an $E$-rational point but $V_1, \ldots, V_k$ do not?
Note that without the condition on subvarieties, every specialization would have the required property.