The strong Fatou–Bieberbach approximation conjecture

Let $f\in{\mathcal{S}}$ be normalized and univalent on ${\mathbb{B}^n}\subseteq{\mathbb{C}^n}$, and suppose that $f({\mathbb{B}^n})$ is not Runge. For $r\in(0,1)$, let $f_r(z)=r^{-1}f(rz)$. Let a Fatou–Bieberbach domain be a domain biholomorphic to ${\mathbb{C}^n}$, and let ${\mathcal{S}}({\mathbb{C}^n})$ denote the normalized entire univalent maps.

Strong Fatou–Bieberbach approximation conjecture. For each $r\in(0,1)$, there exist a Fatou–Bieberbach domain $\Omega_r$ and a sequence $\Psi_k^{(r)}\in{\mathcal{S}}({\mathbb{C}^n})$ such that

$$\Psi_k^{(r)}({\mathbb{C}^n})=\Omega_r$$

for every $k$, and $(\Psi_k^{(r)})$ converges to $f_r$ uniformly on compact subsets of ${\mathbb{B}^n}$. This is presented as another weaker formulation of the strong generalized Andérsen–Lempert conjecture; no resolution is supplied.