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No. If you blow up $\mathbb{P}^n$ along a smooth (closed) subvariety $X$ of codimension $\geq 2$, the Hodge conjecture for the resulting variety is equivalent to the Hodge conjecture for $X$. So the Hodge conjecture for rational varieties (= birational to $\Bbb{P}^n$) implies the Hodge conjecture in general.