The wild McKay point-counting conjecture for quotient singularities

Let $K$ be a local field with finite residue field $\kappa$ of cardinality $q$, let $\tau:\Gamma\to\operatorname{GL}_n\mathcal{O}_K$ be a faithful representation of a finite group $\Gamma$, and let $X:=\mathbb{A}_{\mathcal{O}_K}^n/\Gamma$. Suppose that for every $g\in\Gamma\setminus\{1\}$, the fixed-point locus $(\mathbb{A}_{\mathcal{O}_K}^n)^g$ has codimension at least two, and let $f:Y\to X$ be a crepant resolution. Write $Z\subset Y$ for the preimage of the origin $0\in X(\kappa)$. For a continuous homomorphism $\rho:G_K\to\Gamma$, let $\mathbf{w}_\tau(\rho)$ denote its weight, and let $M(K,\Gamma,-\mathbf{w}_\tau)$ be the corresponding mass.

Point-counting realization of the wild McKay conjecture. If the residue field $\kappa$ is finite, then

$$\sharp Z(\kappa)=M(K,\Gamma,-\mathbf{w}_\tau).$$

This is the finite-residue-field point-counting form of the motivic wild McKay correspondence, relating the special fiber of a crepant resolution to a weighted count of $\Gamma$-torsors over $\operatorname{Spec}K$. It is posed as a consequence of the motivic conjecture and remains unresolved in the stated generality.