The curve selection conjecture for irreducible components of arc spaces

Let $X$ be a complex space, let $W$ be an irreducible component of $\operatorname{Arc}^{\circ}(X)$ or $\operatorname{ShArc}(X)$, and let $[\phi_1],\dots,[\phi_m]$ be arcs in $W$. Curve selection conjecture. There is a holomorphic family of arcs

$$F:{\mathbb D}\times\overline{\mathbb D}\to X$$

and points $p_1,\dots,p_m\in{\mathbb D}$ such that

$$\phi_i(t)=F(p_i,t)$$

for $i=1,\dots,m$. This would provide a curve-selection principle for the infinite-dimensional arc spaces in question. The source describes the assertion as much harder than the corresponding statement for algebraic varieties and does not prove it.