Connectivity conjecture for marked cycle curves in the basilica limb

Let $\mathcal{F}_2$ denote the relevant parameter space and let $\operatorname{Cyc}_p(\mathcal{F}_2)$ be the marked cycle curve of period $p$. Connectivity conjecture. For $p\ne 3$, the marked cycle curve $\operatorname{Cyc}_p(\mathcal{F}_2)$ is connected. The only noted exception is period $3$, where the marked cycle curve is disconnected because the corresponding real primitive cycle does not provide an edge joining the two vertices.