Finite-group quotient structure of fibers of wild jet maps

Let $G$ act linearly on $V=\mathbb{A}_{D}^{d}$, let $p_n:\pi_n(J_\infty^GV)\to J_nX$ be the natural map for $X=V/G$, and let $\mathbf{j}_p$ and $\mathbf{w}_V$ be the Jacobian-order and weight functions. For $\gamma\in J_\infty^GV$, write $(\mathbf{j}_p-\mathbf{w}_V)(\gamma)$ for the corresponding integer.

Fiber-structure conjecture. For $n\gg0$, the fiber of $p_n$ over the image of $\gamma$ is homeomorphic to a quotient of

$$\mathbb{A}_{k}^{(\mathbf{j}_p-\mathbf{w}_V)(\gamma)}$$

by a linear action of a finite group.

This conjecture is presented as the geometric source of the preceding change-of-variables formula. The source gives no resolution.