Characterisation of wild auxiliary arcs between vertices

Let $G$ be a graph and let $x$ and $y$ be distinct vertices of $G$. Write $\overset{{}_{\;\rightarrow}}{S}(G-xy)$ for the partially ordered set of oriented vertex separators of $G-xy$, let $T_2$ be the infinite binary tree, and let $\langle\,\cdot\,\rangle\colon\bigcup_{n<\omega}2^n\to\mathbb{I}$ map each binary sequence to its naturally corresponding value in the unit interval. An auxiliary arc from $x$ to $y$ is called wild when it has the wildness property defined in the paper, and % \mathrel{\vcenter{\offinterlineskip \text{\sim}\vskip-.35ex\text{\sim}\vskip-.35ex\text{\sim}}} and $\sim$ denote the paper's auxiliary equivalence relations. Wild-arc characterisation conjecture. For every graph $G$ and every two distinct vertices $x$ and $y$ of $G$, the following are equivalent: (1) $x\sim y$ and every auxiliary arc from $x$ to $y$ is wild; (2) there \exists a countable \subset $W$ of $\overset{{}_{\;\rightarrow}}{S}(G-xy)$ such that: (a) every $(A,B)\in W$ satisfies $x\in A\setminus B$ and $y\in B\setminus A$; (b) the \partial ordering $\le$ of $\overset{{}_{\;\rightarrow}}{S}(G-xy)$ induces the ordering of the rationals on $W$; and (c) there \exists a bijection $\varphi\colon T_2\to W$ satisfying $\varphi(a)\le\varphi(b)$ if and only if $\langle a\rangle\le\langle b\rangle$. If $\hat{\varphi}\colon T_2\to\mathcal{X}$ maps each vertex of $T_2$ to the vertex separator of $\varphi(a)$, $n(a)\in\mathbb{N}$ is the level containing $a$, and $X_a$ is the set of all vertices in the first $n(a)$ levels of $T_2$ other than $a$, then for every $a\in V(T_2)$ and every auxiliary arc $A$ from $x$ to $y$, the set $\hat{\varphi}(a)\setminus\bigcup_{b\in X_a}\hat{\varphi}(b)$ meets $A$; or (3) x\not% \mathrel{\vcenter{\offinterlineskip \text{\sim}\vskip-.35ex\text{\sim}\vskip-.35ex\text{\sim}}} y.

The conjecture seeks a structural characterisation of when auxiliary arcs are wild, using a countable rationally ordered family of separators arranged like the infinite binary tree. The supplied passage raises this as an outlook question and gives no evidence of resolution, so its status is open.