Locally dense wandering Fatou components with high emergence

Let $\operatorname{Aut}(\mathbb C^2)$ denote the space of polynomial automorphisms of $\mathbb C^2$. A dynamical system has a wandering Fatou component when it displays a Fatou component whose forward images are pairwise distinct. Locally dense wandering-component conjecture. There is a locally dense set in the space of polynomial automorphisms $\operatorname{Aut}(\mathbb C^2)$ of $\mathbb C^2$ formed by dynamics displaying a wandering Fatou component with high emergence. The preceding results establish related real and surface-diffeomorphism phenomena, but the proposed statement for polynomial automorphisms of $\mathbb C^2$ is presented as a possible direction for extending the techniques and remains open.