The conjectural change-of-variables formula for linear wild McKay correspondence

Let $X=V/G$ for a linear action of a finite group $G$ on $V=\mathbb{A}_{D}^{d}$. Let $p_\infty:J_\infty^GV\to J_\infty X$ be the natural map, let $\mathbf{j}_p$ be the Jacobian-order function, and let $\mathbf{w}_V$ be the weight function. For a measurable function $\Phi:J_\infty X\supset C\to\mathcal{R}\cup\{\infty\}$, let $p_\infty^{-1}(C)$ be the inverse image.

Linear wild change-of-variables conjecture.

$$\int_C\Phi\,d\mu_{J_\infty X}=\int_{p_\infty^{-1}(C)}(\Phi\circ p_\infty)\mathbb{L}^{-\mathbf{j}_p+\mathbf{w}_V}\,d\mu_{J_\infty^GV}.$$

This formula is the central conjectural change-of-variables statement from which the linear motivic McKay correspondence follows. The source indicates that it is conjectural and gives no resolution.