The weak virtual periodicity conjecture for maximal coclass trees

Let $r\in\mathbb N_0$ and let $K$ be an arbitrary field. For an algebra $A$ in $\mathcal G_{r,K}$, write $\mathcal T(A)$ for the rooted subgraph consisting of all paths starting at $A$. A maximal coclass tree is a coclass tree containing a unique maximal infinite path and satisfying the maximality condition from the source. Let $\mathcal T$ be such a tree, with maximal infinite path $A_1\to A_2\to\cdots$, and let $\overline{\mathcal T}(A)$ denote its unlabelled version rooted at $A$. Weak virtual periodicity conjecture. There exist positive integers $l$ and $k$ such that

$$\overline{\mathcal T}(A_l)\cong\overline{\mathcal T}(A_{l+k}).$$

This conjecture predicts eventual periodicity of the unlabelled maximal coclass trees and would imply finite width. Its resolution is not supplied here.