The strong virtual periodicity conjecture for maximal coclass trees

Let $r\in\mathbb N_0$ and let $K$ be an arbitrary field. Let $\mathcal T$ be a maximal coclass tree in $\mathcal G_{r,K}$ with maximal infinite path $A_1\to A_2\to\cdots$. Write $\overline{\mathcal T}(A)$ for the unlabelled rooted tree at $A$. Strong virtual periodicity conjecture. There exist positive integers $l$ and $k$, a graph isomorphism

$$\mu:\overline{\mathcal T}(A_l)\longrightarrow\overline{\mathcal T}(A_{l+k}),$$

and, for every vertex $B\in\mathcal T(A_l)\setminus\mathcal T(A_{l+k})$, a rational polynomial $f_B$ such that the label of $\mu^i(B)$ equals $f_B(i)$. This strengthens weak virtual periodicity by requiring eventual polynomial behaviour of labels; its resolution is not supplied here.