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Let $S$ be a scheme in characteristic $0$, and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme $X$ by a stack $\mathscr{X}$.
Does $\mathscr{X}$ always admit an atlas $Y$ over $S$ which can be enhanced to a crystal of schemes over $S$? In other words, is there a crystal $Y$ in schemes over $S$ together with a map of algebraic stacks $Y\to \mathscr{X}$ which is smooth and surjective?