Internal characterization of non-horocyclically convex domains in the unit disk

Let $\mathbb{D}$ denote the unit disk, and let $G$ be a simply connected subdomain of $\mathbb{D}$. Let $\mathcal{C}$ be the class of domains $\Omega\subset\mathbb{D}$ of the form $D_1\setminus\overline{D_2}$, where each $D_j$ is a hyperbolic disk in $\mathbb{D}$ and $\partial D_1$ and $\partial D_2$ intersect orthogonally at two points in $\mathbb{D}$. For $\Omega\in\mathcal{C}$, let $m(\Omega)$ denote the hyperbolic midpoint of the hyperbolically concave boundary arc $\partial D_2\cap D_1$ of $\Omega$. Internal characterization conjecture. $G$ is not horocyclically convex in $\mathbb{D}$ if and only if there is a domain $\Omega\in\mathcal{C}$ such that $\Omega\subset G$ and $m(\Omega)\in\partial G\cap\mathbb{D}$. The paper presents this as a proposed characterization motivated by known results on lower bounds for the hyperbolic metric, but states that the related questions have no answers in the current work.