Twin-period recurrence conjecture for reversible elementary cellular automaton 115

Let reversible elementary second-order cellular automaton 115 (RESOCA 115) have finite configurations with periods. Two periods $p$ and $p+3$ are called twins.

Twin-period recurrence conjecture. Assume that $p$ and $p+3$ are twin periods. Then the next twin pair is $p'$ and $p'+3$, where

$$p'=2p+\frac{9-3(-1)^p}{2}.$$

This conjecture proposes a recurrence for the experimentally observed twin periods of finite configurations of RESOCA 115; the source states that the authors do not know whether all periods are twins and presents the recurrence as a pattern suggested by experiments.