Skewer conjecture for marked cycle curves

Let $\pi$ be the projection from the marked cycle curve $\operatorname{Cyc}_p(\mathcal{F}_1)$ to its parameter space, and consider the lift of the real axis under this projection. Skewer conjecture. For every $p\geq 3$, the closure of the lift of the real axis under $\pi$ is connected in $\operatorname{Cyc}_p(\mathcal{F}_1)$. This is a strengthening of the known connectedness of the Riemann surface $\operatorname{Cyc}_p(\mathcal{F}_1)$.