Existence of the moduli space of twisted arcs

Let $\mathcal{X}$ be a Deligne–Mumford stack of finite type over $D$. A stacky formal disk is a connected normal Deligne–Mumford stack birational and finite over $D$, and a twisted arc of $\mathcal{X}$ is a representable $D$-morphism $\mathcal{E}\to\mathcal{X}$ from a stacky formal disk. Twisted-arc moduli conjecture. There exists a moduli space parameterizing isomorphism classes of twisted arcs of $\mathcal{X}$, denoted by $\mathcal{J}_{\infty}\mathcal{X}$. Such a space is needed to formulate motivic integration on wild Deligne–Mumford stacks; its existence is assumed subsequently and remains open in the paper.