Connectedness conjecture for uni-Julia sets of coupled logistic-map networks

Let one of the three network models described in the paper be fixed, let $c\in\mathbb{C}$ be the equi-parameter, and denote the corresponding uni-J set and equi-M set by the network's uni-J and equi-M sets.

Connectedness conjecture. For any of the three models described, and for any equi-parameter $c\in\mathbb{C}$, the uni-J set is connected only if $c$ is in the equi-M set of the network, and it is totally disconnected only if $c$ is not in the equi-M set of the network.

This conjecture proposes a connectedness dichotomy for uni-J sets in the three coupled logistic-map models, while allowing intermediate cases with finitely or infinitely many connected components that are not totally disconnected. Its status is not established by the supplied source context.