Conjecture relating Nash adjacency to the thresholds $k_E$ and $k_F$

Let $E$ and $F$ be prime divisors with contact order

$$\operatorname{Cont}(E,F)=i,$$

where the points $p^E_{i+1}$ and $p^F_{i+1}$ are both free. Define

$$k_F:=\max\left\{m:\bigcup_{F'\equiv_i F}\overline{N}_{F'}\subsetneq\overline{\operatorname{Cont}^{m}(E_{p_i})}\right\},$$

and

$$k_E:=\min\left\{m:\overline{\operatorname{Cont}^{m}(E_{p_i})}\subsetneq\bigcap_{E'\equiv_{\geq i}E}\overline{N}_{E'}\right\}.$$

Here $\overline{N}_E$ denotes the Nash set associated with $E$, and $\overline{\operatorname{Cont}^{m}(E_{p_i})}$ is the corresponding closed maximal divisorial set.

Threshold conjecture. One has

$$\overline{N}_F\subsetneq\overline{N}_E \quad\Longleftrightarrow\quad k_F\geq k_E.$$

The conjecture seeks a numerical criterion for Nash adjacency in the case where the next points in the resolutions are free. Computing $k_E$ and $k_F$ is also left as an open problem, and the parser supplies no evidence that the equivalence has been resolved.