The motivic McKay correspondence for non-linear actions

Let $\mathfrak{x}$ be the centered log structure on the quotient $\mathsf{x}:=\mathsf{v}/G$, and let $\mathfrak{v}^{|F|,\nu}$ be the induced centered log structure on the normalization of $\mathsf{v}^{|F|}$, where $F$ is a connected component of a $G$-cover $E\to D$ and $C_G(H)$ is the centralizer of its stabilizer $H$. Let $M_{\mathrm{st},C_G(H)}(\mathfrak{v}^{|F|,\nu})$ denote the corresponding equivariant stringy motif, and let $\tau$ be the tautological motivic measure on the conjectural moduli space $G\text{-}\mathrm{Cov}(D)$.

The motivic McKay correspondence for non-linear actions. We have

$$M_{\mathrm{st}}(\mathfrak{x})=\int_{G\text{-}\mathrm{Cov}(D)}M_{\mathrm{st},C_G(H)}(\mathfrak{v}^{|F|,\nu})\,d\tau.$$

This generalizes the linear wild McKay correspondence to non-linear actions on possibly singular affine varieties. Its formulation depends on the conjectural moduli spaces and motivic measures, and the source gives no resolution.