Existence and truncation properties of twisted jet spaces

Let $\mathcal{X}$ be a Deligne–Mumford stack of finite type over $D$, and let $\mathcal{J}_{\infty}\mathcal{X}$ be its moduli space of twisted arcs. For $n\in\mathbb{Z}_{\ge0}$, a twisted $n$-jet is the corresponding finite-level truncation of a twisted arc. Twisted-jet conjecture. For each $n\in\mathbb{Z}_{\ge0}$, there exists a moduli space $\mathcal{J}_n\mathcal{X}$ of twisted $n$-jets such that, if $\mathcal{X}=[M/G]$, then $\mathcal{J}_n\mathcal{X}=J_n^GM$, and there are truncation maps $\mathcal{J}_{n+1}\mathcal{X}\to\mathcal{J}_n\mathcal{X}$ making $\mathcal{J}_{\infty}\mathcal{X}$ the projective limit of the $\mathcal{J}_n\mathcal{X}$. This would extend the familiar jet-space tower to Deligne–Mumford stacks, but the existence and stated compatibility are conjectural here.