Reduced auto-arc affine-space criterion for smoothness

Let $X$ be a variety over $k$, let $p$ be a point of $X$, and let $\mathcal{A}_n(X,p)$ be its auto-arc space at level $n$. Let $\kappa(p)$ be the residue field at $p$, and write $\mathcal{A}_n(X,p)^{\operatorname{red}}$ for the reduction. Auto-arc affine-space smoothness conjecture. If, for all sufficiently large $n$, there is an $r_n\in\mathbb{N}$ such that

$$\mathcal{A}_n(X,p)^{\operatorname{red}}\cong\mathbb{A}_{\kappa(p)}^{r_n},$$

then $X$ is smooth at $p$. The preceding theorem proves the corresponding local auto-arc description at smooth points; the converse criterion is proposed as a way to detect smoothness.