The reductive local structure conjecture for algebraic stacks

Let $\mathcal{X}$ be a quasi-separated algebraic stack, locally of finite type over a noetherian scheme, with affine automorphism groups. Let $x\in\mathcal{X}$ be a closed point with reductive stabilizer. Then there exists an étale morphism

$$(\operatorname{Spec} A/\operatorname{GL}_n,w)\to(\mathcal{X},x)$$

that induces an isomorphism of automorphism groups at $w$.

Reductive local structure conjecture. Under these hypotheses, such an étale morphism exists.

This conjecture proposes that the second local structure theorem extends to positive characteristic despite the failure of linear reductivity for groups such as $\operatorname{GL}_n$. The supplied text says that the result is reasonable to hope for but gives no resolution evidence.