Is it true that for any DM stack $\mathcal{X}$ (quasicompact, separated, of finite type over a field) there is a Zariski covering $\mathcal{U}_i \to \mathcal{X}$ by open substacks, such that for all $i$ there is an etale finite surjective morphism $\phi_i: Z_i \to \mathcal{U}_i$, where $Z_i$ is a scheme? (Of course it will be true if we remove the condition that the morphisms $\phi_i$ must be finite.)

Upd. Oh, I think by Theorem 6.1 of Laumon, Moret-Bailly "Champs algebriques" it is equivalent to saying that $\mathcal{X}$ admits a Zariski covering by quotient stacks.