The neighborhood-independence conjecture for arc spaces of isolated singularities

Let $(0\in X)$ be an isolated singularity, and let $0\in U\subset X$ be a contractible, Stein neighborhood. Neighborhood-independence conjecture. The homeomorphism types of $\operatorname{Arc}(U)$ and $\operatorname{Arc}^{\circ}(U)$ are independent of $U$. A positive answer would give a suitable replacement for the failure of the homeomorphism type of $\operatorname{Arc}^{\circ}(X)$ to be invariant under changing the representative of a pointed singularity. The source presents this as an open problem and gives no proof.