The Fatou–Bieberbach realization conjecture for non-Runge ball domains

Let $D\subseteq{\mathbb{C}^n}$ be a domain biholomorphic to the ball ${\mathbb{B}^n}$, and suppose that $D$ is not Runge in ${\mathbb{C}^n}$. A Fatou–Bieberbach domain is a proper domain in ${\mathbb{C}^n}$ biholomorphic to ${\mathbb{C}^n}$; a pair $(D,\Omega)$ is Runge when $D$ is Runge in $\Omega$.

Fatou–Bieberbach realization conjecture. There exists a Fatou–Bieberbach domain $\Omega\subseteq{\mathbb{C}^n}$ with $D\subseteq\Omega$ such that $(D,\Omega)$ is a Runge pair. This is proposed as a necessary condition for embedding all normalized univalent maps into Loewner chains, and its status is not resolved in the supplied text.