Periodicity conjecture for the twigs of frame groups

Let $p\geq 5$ be prime. For a group $G$ in the frame $\mathcal F_i$, let $\mathcal R(G)$ denote the twig: the subtree of $\mathcal B_i$ consisting of the descendants of $G$ that are not in the frame. For each $i$ and $\gamma\in\hat{H}_i$, the groups $S_{i,m}(\gamma)$ are the groups used in the frame construction.

Twig periodicity conjecture. There exist integers $e=e(p)$ and $f=f(p)$ with $(p-1)\mid f$ such that, for every $i\geq e$ and every $\gamma\in\hat{H}_i$,

$$\mathcal R\bigl(S_{i+f,m}(\gamma)\bigr)\cong\mathcal R\bigl(S_{i,m}(\gamma)\bigr).$$

This is presented as a variation of Conjecture W of Eick, Leedham-Green, Newman, and O'Brien. The paper gives no resolution, so the asserted eventual periodicity remains open.