The eventual special-vertex conjecture for Hanoi border paths

Let $p\geq 3$, let $G^{(p)}_{n+1}$ contain the copy $AG^{(p)}_n$, and let $S_{n_0}$ be the recursively defined border set. Let $a$ be the vertex labeled $A^{n_0+1}$, and call a vertex special when its label has the form $A^{n-i}X^i$ with $X\in{\cal A}_p\setminus\{A,B\}$ in the relevant shortest-path setting. Eventual special-vertex conjecture. For any $p\geq 3$, there is $n_0\geq 2$ such that for any $v\in S_{n_0}$ there is a shortest path from $a$ to $v$ that meets a special vertex. This is proposed as the inductive base needed for the partial proof of the special-vertex result; the supplied text gives no resolution evidence.