Characterization of complementary components in the closure of linearly accessible domains

Determine whether every connected component Q of the complement of the union of the limiting half-lines and the domain, namely Q a connected component of (C\setminus\Omega)\setminus\mathcal L with \mathcal L=\bigcup_{w\in\partial\Omega}L_w, is necessarily an angular set whose vertex belongs to \partial\Omega.

Background

Appendix B studies the stability of linearly accessible schlicht functions under locally uniform convergence. Given domains \Omega_n=f_n(D) converging locally uniformly to \Omega=f(D), the paper constructs, for every boundary point w\in\partial\Omega, a limiting half-line L_w contained in C\setminus\Omega, with the family of half-lines having no transversal crossings.

To complete the proof that the limiting domain \Omega remains linearly accessible, the authors consider the connected components Q of the portion of C\setminus\Omega not covered by the limiting half-lines. The unresolved geometric step is whether each such component must be an angular set with its vertex on the boundary of \Omega; establishing this would make the remainder of the closedness argument straightforward.