Visibility characterization by cluster sets of non-visible geodesics

Let $\Omega\subset\mathbb C^N$ be a bounded convex domain. Let $\{\gamma_k\}$ be a sequence of non-visible geodesic segments and define

$$\Gamma(\{\gamma_k\}):=\{p\in\partial\Omega: \exists\{t_{k_n}\}\subset[0,1]\text{ such that }\lim_{n\to\infty}\gamma_{k_n}(t_{k_n})=p\}.$$

The cluster-set conjecture. If $p,q\in\Gamma(\{\gamma_k\})$ and $p\ne q$, then either

$$\{tp+(1-t)q:0<t<1\}\subset\Omega,$$

or

$$\{tp+(1-t)q:0\le t\le1\}\subset\partial\Omega$$

and

$$(\mathbb C(p-q)+q)\cap\Omega=\emptyset.$$

This describes the possible boundary cluster sets of non-visible geodesic segments and is intended to clarify the geometric obstruction to visibility in bounded convex domains; the source provides no resolution.