Brain teaser: generic perversity on fibers

Inspired by Shizhang’s Rampage talk last week, here is a brain teaser. Feel free to post your solution in the comments!

Let $f:X \to Y$ be any map of irreducible complex varieties, and let $\mathcal{F}$ be a perverse sheaf on $X$ . Prove that there is a dense open subset $U \subset Y$ such that for any closed point $y \in U$ , the shifted restriction $(\mathcal{F}|X_y)[-\dim Y]$ is a perverse sheaf on the fiber $X_y$ .

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