The conjectural change-of-variables formula for untwisting

Let the assumptions of the untwisting construction hold. Let $X=V/G$, let $E\to D$ be a $G$-cover with connected component $F$, and let $p^{|F|}:V^{|F|}\to X$ be the natural morphism. Let $A\subset p_\infty(J_\infty^{G,E}V)$ and let $\Phi:J_\infty X\supset A\to\mathcal{R}\cup\{\infty\}$ be measurable. Write $p_{(\infty)}^{|F|}:(J_\infty V^{|F|})/C_G(H)\to J_\infty X$ for the induced map, and let $\mathbf{j}_{p^{|F|}}$ be its Jacobian-order function.

Untwisted change-of-variables conjecture.

$$\int_A\Phi\,d\mu_{J_\infty X}=\int_{(p_{(\infty)}^{|F|})^{-1}(A)}(\Phi\circ p_{(\infty)}^{|F|})\mathbb{L}^{-\mathbf{j}_{p^{|F|}}}\,d\mu_{(J_\infty V^{|F|})/C_G(H)}.$$

This is a conjectural change-of-variables formula for the untwisting map and is intended to derive the linear McKay correspondence. The source gives no resolution.