Conjecture on intersections of Nash sets and maximal divisorial sets

Let $E$ be a prime divisor, and let $\{p_i\}_{i\in I}$ be the infinitely near points blown up to obtain the minimal model of $E$. Let $p_{i_0}$ be any free point among them. Write $E_{p_{i_0}}$ for the divisor associated with $p_{i_0}$, and let $\overline{\operatorname{Cont}^q(E_{p_{i_0}})}$ denote the closure of the maximal divisorial set at order $q$.

Intersection conjecture. For some $q$, the intersection

$$\bigcap_{E'\equiv_{i_0} E}\overline{N}_{E'}$$

coincides with

$$\overline{\operatorname{Cont}^q(E_{p_{i_0}})}.$$

The conjecture concerns the relation between Nash sets and maximal divisorial sets. The paper leaves both the existence and the value of $q$ conjectural; in the special one-Puiseux-pair case, a related equality is known, but the stated equality is false in general for the analogous $i_0=1$ formulation.