The étale-local quotient stack conjecture

Let $X$ be an algebraic stack locally of finite type over an algebraically closed field $k$ with separated and quasi-compact diagonal, and suppose that $X$ has affine stabilizer groups at all closed points. Let $x\in X(k)$ have linearly reductive stabilizer group scheme $G_x$. The étale-local quotient stack conjecture. There exists an affine scheme $W$ over $k$ with an action of $G_x$, a $k$-point $w\in W$, and an étale, representable morphism

$$f\colon [W/G_x]\to X$$

such that $f(w)=x$ and $f$ induces an isomorphism of stabilizer groups at $w$. The conjecture asserts that algebraic stacks are étale locally quotient stacks around points with linearly reductive stabilizers; the paper proves this statement for the moduli stack of pointed prestable curves, while the general claim remains open in the supplied context.