Three-periodicity conjecture for lower central quotients of the groups Γ_T

Let $\Gamma=\Gamma_{\mathcal T}$ be one of the groups introduced in Section 4 of the cited source, associated to a prime power $q=p^3$ with $p\neq 3$. Let $\gamma_i(\Gamma)$ denote the terms of its lower central series. Three-periodicity conjecture. For every $i\geq 2$,

$$\log_p[\gamma_i(\Gamma):\gamma_{i+1}(\Gamma)]= \begin{cases} 3,&\text{if }i\equiv 0,1\pmod 3,\\ 2,&\text{if }i\equiv 2\pmod 3. \end{cases}$$

This predicts a periodic pattern for the ranks of the lower-central-series abelian quotients of the groups associated with prime powers $p^3$, excluding $p=3$ to avoid torsion phenomena. The supplied text presents it as an expectation based on computer calculations and gives no resolution.