The Fatou–Bieberbach approximation conjecture for restrictions of univalent maps

Let $f\in{\mathcal{S}}$ be normalized and univalent on ${\mathbb{B}^n}\subseteq{\mathbb{C}^n}$. For $r\in(0,1)$, write $f_r(z)=r^{-1}f(rz)$. A Fatou–Bieberbach domain is a domain biholomorphic to ${\mathbb{C}^n}$, and a pair $(U,\Omega)$ is Runge when $U$ is Runge in $\Omega$.

Fatou–Bieberbach approximation conjecture. For every $r\in(0,1)$, there exists a domain $\Omega_r\subseteq{\mathbb{C}^n}$ biholomorphic to ${\mathbb{C}^n}$ such that

$$f(r{\mathbb{B}^n})\subseteq\Omega_r$$

and $(f(r{\mathbb{B}^n}),\Omega_r)$ is a Runge pair. This is given as a weaker formulation related to the preceding Fatou–Bieberbach conjectures; its status is not resolved in the supplied text.