The wild McKay correspondence conjecture for linear actions

Let $D=\operatorname{Spec}\mathcal{O}_{D}$, where $\mathcal{O}_{D}$ is a complete discrete valuation ring with algebraically closed residue field $k$. Let $V=\mathbb{A}_{D}^{d}$ carry a linear action of a finite group $G$, and set $X:=V/G$. Let $o\in X(k)$ be the image of the origin, let $M_{\mathrm{st}}(X)_{o}$ be the stringy motif at $o$, and let $G\text{-}\mathrm{Cov}(D)$ denote the conjectural moduli space of $G$-covers of $D$. Let $\mathbf{w}$ be the weight function associated to the representation $V$, and let $\tau$ be the tautological motivic measure.

The wild McKay correspondence conjecture. If the quotient morphism $V\to X$ is étale in codimension one, then

$$M_{\mathrm{st}}(X)_{o}=\int_{G\text{-}\mathrm{Cov}(D)}\mathbb{L}^{\mathbf{w}}\,d\tau.$$

This is the stringy-invariant formulation of the wild McKay correspondence, extending the characteristic-zero and tame cases to arbitrary characteristics. The moduli space and motivic integral are conjectural, and the formula is known only in special cases.