One-singularity curve auto-arc structure conjecture

Let $C$ be a connected reduced curve over $k$ with exactly one singular point $p$, and let $\bar C\to C$ be its normalization of degree $e$. Let $\mathfrak{l}_{P_i(n)}$ denote the linear arc associated with the polynomial $P_i(n)\in\mathbb{Z}[t]$. One-singularity curve auto-arc structure conjecture. For all sufficiently large $n$, there exist polynomials $P_i(t)\in\mathbb{Z}[t]$, with $\deg(P_i(t))\leq1$ for $i=1,\ldots,e$, a fixed $r\in\mathbb{N}$, and a connected curve $W$ analytically isomorphic to $C$ at $O$, such that

$$\mathcal{A}_n(C,p)^{\operatorname{red}}\cong\nabla_{\mathfrak{l}_{P_1(n)}}W\times_k\cdots\times_k\nabla_{\mathfrak{l}_{P_e(n)}}W\times_k\mathbb{A}_k^r.$$

This is the paper's general structural conjecture for curves with one singularity, motivated by the explicit cusp and node calculations; no proof or resolution is given in the supplied text.