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.