The itinerary-preservation conjecture for marked cycles in the basilica degeneration

Let $g_a$ be the family described above, let $C_{p,2}$ denote the set of period-$p$ itineraries, and let

$$\mathcal{I}_a=\{I_a^{(k)}\},\qquad \mathcal{J}=\{J^{(k)}\}$$

be the indicated partitions of the Julia sets for $a<-1$ and $a=0$, respectively. Itinerary-preservation conjecture. Given a periodic cycle of period $p\geq 3$ for $g_a$ with $a<-1$ and itinerary $\omega\in C_{p,2}$ with respect to the partition $\mathcal{I}_a$, analytically continuing the cycle across the arc $[a,0]$ in parameter space yields a periodic cycle of period $p$ for $g_0$ whose itinerary with respect to $\mathcal{J}$ equals $\omega$. This predicts that the symbolic coding is preserved during the continuation from the external component to the basilica component.