Existence of moduli spaces of G-arcs and G-jets

Let $D=\operatorname{Spec}A$ be as above, let $M$ be a $D$-variety endowed with an action of a finite group $G$, and let a G-arc be a $G$-equivariant $D$-morphism $E\to M$ from a $G$-cover $E\to D$. For each $n\in\mathbb{Z}_{\ge0}$, a G-$n$-jet is a pair $(E,E_n\to M)$ consisting of a $G$-cover and a $G$-equivariant morphism from its length-$1+ne$ truncation $E_n$ to $M$. Moduli-space conjecture for G-arcs and G-jets. There exist moduli spaces parameterizing isomorphism classes of $G$-arcs and $G$-$n$-jets of $M$ for every $n$, denoted by $J_{\infty}^{G}M$ and $J_n^{G}M$, respectively. These spaces would generalize ordinary arc and jet spaces to equivariant arcs over possibly wildly ramified covers; their construction is not established in the paper.