The weight-function and change-of-variables conjecture for wild Deligne–Mumford stacks

Let $D=\operatorname{Spec}A$ and let $\mathcal{X}$ be a Deligne–Mumford stack of finite type and pure dimension over $D$, with twisted-arc space $\mathcal{J}_{\infty}\mathcal{X}$. For a point $x\in\mathcal{X}$, let $\operatorname{Aut}(x)$ denote its automorphism group. Main conjecture. For each such $\mathcal{X}$, there exists a canonical function

$$w_{\mathcal{X}}:\mathcal{J}_{\infty}\mathcal{X}\to\left(\bigcup_{x\in\mathcal{X}}\frac{1}{\sharp\operatorname{Aut}(x)}\mathbb{Z}\right)\cup\{\infty\}$$

with $w_{\mathcal{X}}\equiv0$ when $\mathcal{X}$ is a scheme. Moreover, for every proper birational morphism $f:\mathcal{Y}\to\mathcal{X}$ of such stacks and every measurable function $F:\mathcal{J}_{\infty}\mathcal{X}\supset C\to\frac{1}{r}\mathbb{Z}\cup\{\infty\}$, one has

$$\int_{C}\mathbb{L}^{F+w_{\mathcal{X}}}d\mu_{\mathcal{X}}=\int_{f_{\infty}^{-1}(C)}\mathbb{L}^{F\circ f_{\infty}-\operatorname{ord}\,\operatorname{Jac}_{f}+w_{\mathcal{Y}}}d\mu_{\mathcal{Y}}.$$

This is the central extension of motivic change of variables to wild stacks. The conjecture also supplies the canonical weight correction, which vanishes for schemes; it remains unproved.