Disjointness conjecture for components of the embedded Nash space

Let $X$ be an integral separated surface, $C$ an effective Cartier divisor in $X$, and $o$ a closed point in $C$. Assume that $X$ is smooth away from $C$. Write $\mathcal{X}_m(X,C,o)$ for the embedded Nash space associated with $(X,C,o)$. Disjointness conjecture. The irreducible components of $\mathcal{X}_m(X,C,o)$ are disjoint. This extends the corresponding result for smooth ambient surfaces, where the disjointness of the irreducible components is known; the conjecture asks whether it remains true when $X$ is singular along $C$.