Twin conjecture for periodic branches of groups of maximal class

Let $p\geqslant 5$ and write $d=p-1$. For sufficiently large $n$, consider the branch $\mathcal B_p(n)$ and its pruned subtrees, and call two groups twins if their descendant trees are isomorphic, that is, $\mathcal D(G)\cong\mathcal D(H)$. The notation $\mathcal B_p(n,k)$ denotes the subtree of groups in $\mathcal B_p(n)$ at depth at most $k$, while $\mathcal D(G,m)$ denotes the corresponding pruned descendant tree.

Twin conjecture. There exists $n_0=n_0(p)$ such that for all $n\geqslant n_0$ and each group $G$ at depth $n-c$ in $\mathcal B_p(n+d)$, there exists a twin $H$ at depth $n-c-d$ in $\mathcal B_p(n)$.

This conjecture concerns the second periodicity of the graphs of $p$-groups of maximal class beyond the already established first periodicity of their shallow pruned branches. It is described as a main open problem and a driving force of current research.