The natural atlas conjecture for nonsingular arcs

Let $X$ be a complex space, and let $\operatorname{Arc}^*(X)\subset \operatorname{Arc}(X)$ be the space of arcs that are not contained in $\operatorname{Sing} X$. A finite type family of arcs is a family parametrized by an analytic space together with finitely many holomorphic-disc parameters, as defined in the paper. Natural atlas conjecture. $\operatorname{Arc}^*(X)$ has a natural atlas consisting of finite type families. The authors leave the precise allowed coordinate changes vague; methods in the cited literature seem close to a solution, but the conjecture is not established in the source.