Exponential synchronization conjecture for circular automata of composite order

Let $\mathcal{A}_n(\mathbf{a},\mathbf{b})$ be a circular automaton of order $n$, where $\mathbf{a}:\mathbb{Z}_n\to\mathbb{Z}_n$ is fixed and $\mathbf{b}\in\mathcal{M}_n$ is chosen from the relevant set of maps. Exponential synchronization conjecture. There exists a constant $\alpha$ with $0<\alpha<1$ such that, as $n\to\infty$,

$$\operatorname{Pr}\left\{\mathbf{b}\in\mathcal{M}_n:\mathcal{A}_n(\mathbf{b})\text{ synchronizes}\right\}=1-O(\alpha^n).$$

The conjecture is proposed as a weaker extension of the paper's decay-rate result from prime order to circular automata of composite order. Its resolution is not stated in the supplied text.