Extension of the longest-period distribution theorem to $\sigma>r$

Let $\sigma$ and $r$ be fixed positive integers, with $\sigma>r$, and let $X_{\sigma,n}$ denote the longest temporal period of a periodic solution for the random-rule cellular automaton described in the paper. Then the normalized variable

$$\frac{X_{\sigma,n}}{n^{\sigma/2}}$$

converges in distribution as $n\to\infty$.

Extension conjecture. Theorem 1 holds in the same form for $\sigma>r$; in particular, for any fixed $\sigma\geq 1$ and $r\geq 1$ with $\sigma>r$, $X_{\sigma,n}/n^{\sigma/2}$ converges in distribution.

The conjecture extends the rigorously established longest-period result beyond the regime where independence among arcs in the directed escape cycle fails. The paper reports that simulations for $\sigma>r$ support the same limiting behavior, but a rigorous analysis remains open.