The special-vertex shortest-path conjecture for Hanoi graphs

Let $p\geq 3$, let ${\cal A}_p$ be an alphabet of $p$ letters, and let $A,B\in{\cal A}_p$. For $n\geq 2$, consider vertices labeled $A^n$ and $B^n$. A vertex on a path is special if its label has the form $A^{n-i}X^i$ for some $X\in{\cal A}_p\setminus\{A,B\}$. Special-vertex conjecture. There is a shortest path with at least one special vertex on every path between vertices $a$ and $b$ with labels $A^n$ and $B^n$. The statement is proved in the source for $p=4$ as a lemma, while the text conjectures that the same technique should apply more generally; the supplied parser gives no resolution evidence for the conjectural general statement.