Factorization of the weight function for quotient stacks

Let $D=\operatorname{Spec}A$, let $G$ be a finite group, and let $V$ be a $G$-representation over $D$. Put $\mathcal{X}=[V/G]$, and identify its twisted-arc space with the equivariant arc space $J_{\infty}^{G}V$. Let $\operatorname{GCov}D$ be the moduli space of unpointed $G$-covers and let $w_V:\operatorname{GCov}D\to\mathbb{Q}$ be the weight function defined from the representation. Weight-factorization conjecture. The weight function $w_{\mathcal{X}}$ factors as

$$w_{\mathcal{X}}:\mathcal{J}_{\infty}\mathcal{X}=J_{\infty}^{G}V\to\operatorname{GCov}D\xrightarrow{w_V}\mathbb{Q}.$$

Thus the stack weight depends only on the underlying $G$-cover and the representation. This is proposed as a concrete description of the abstract weight function in the quotient-stack case and remains conjectural.