Existence of moduli spaces of pointed and unpointed G-covers

Let $A$ be a complete discrete valuation ring with algebraically closed residue field, let $D=\operatorname{Spec}A$, and let $G$ be a finite group. A $G$-cover of $D$ is the finite cover associated with an étale $G$-torsor over the punctured disk $D^*$. A pointed $G$-cover additionally specifies a connected component. Moduli-space conjecture for G-covers. There exist moduli spaces parameterizing isomorphism classes of pointed and unpointed $G$-covers of $D$, denoted by $\operatorname{GCovPt}D$ and $\operatorname{GCov}D$, respectively. These spaces are intended to provide the parameter spaces used in motivic integration over wild stacks; their precise construction is left open.