Alper's étale-local quotient conjecture for Artin stacks

Let $\mathcal{X}$ be an Artin stack finitely presented over an algebraic space $S$, and let $\xi\in|\mathcal{X}|$ be a closed point with linearly reductive stabilizer and image $s\in S$.

Alper's étale-local quotient conjecture. There exists an étale neighborhood $S'\to S$ with $s'\mapsto s$ and an étale, representable morphism

$$f:[X/G]\to\mathcal{X}$$

where $G\to S'$ is a flat, finitely presented group algebraic space acting on an algebraic space $X\to S'$, such that for any geometric point $x\in\mathcal{X}(k)$ representing $\xi$, the groups $\operatorname{Aut}_{\mathcal{X}(k)}(x)$ and $G\times_{S'}k$ are isomorphic. Moreover, there is a lift of $\xi$ to $x:\operatorname{Spec}k\to X$ such that $f$ induces an isomorphism

$$G_x\to\operatorname{Aut}_{\mathcal{X}(k)}(f(x)).$$

This conjecture asks whether the formal local quotient structure for Artin stacks with linearly reductive stabilizers extends to the étale topology. A positive answer would support intrinsic constructions of moduli spaces for objects with infinite automorphisms; the paper establishes the corresponding formal-local result and provides evidence, including algebraicity of the stabilizer action on miniversal deformation spaces.