The special-stabilizer lifting conjecture for arcs of Artin stacks

Let $k$ be an algebraically closed field of characteristic $0$. Let $\mathcal{X}$ be a finite type Artin stack over $k$ admitting a good moduli space $\pi: \mathcal{X} \to X$, where $X$ is an irreducible $k$-scheme and $\pi$ induces an isomorphism over a nonempty open subset of $X$. Let $\operatorname{sep}_\pi: \mathscr{L}(X) \to \mathbb{N}$ be the function measuring the non-separatedness of $\pi$, and let $\mu_X$ be the usual motivic measure on $\mathscr{L}(X)$. The stabilizers of $\mathcal{X}$ are special if they are special groups in the sense of Serre.

Special-stabilizer lifting conjecture. If all stabilizers of $\mathcal{X}$ are special groups, then $\operatorname{sep}_\pi^{-1}(0) \subset \mathscr{L}(X)$ is measurable and

$$\mu_X\bigl(\operatorname{sep}_\pi^{-1}(0)\bigr)=0.$$

Equivalently, almost all arcs of $X$ lift to finitely many arcs of $\mathcal{X}$. This conjecture would make it possible to compute stringy invariants through ordinary arcs on suitable Artin stacks; the paper establishes the surrounding framework and verifies its conjectures for fantastacks, but does not establish this assertion in the stated generality.