Abate's full omega-limit conjecture for self-intersecting geodesics

Let $S^o=\mathbb{P}^1(\mathbb{C})\setminus\{p_0,p_1,\dots,p_r\}$, where $p_0,p_1,\dots,p_r$ are the poles of a meromorphic connection $\nabla$ on $\mathbb{P}^1(\mathbb{C})$. Let $\sigma:[0,\varepsilon)\to S^o$ be a maximal geodesic. Abate's conjecture. If $\sigma$ intersects itself infinitely many times, then the $\omega$-limit set of $\sigma$ is $\mathbb{P}^1(\mathbb{C})$. This conjecture concerns the unresolved behavior of geodesics with infinitely many self-intersections; earlier work classified omega-limit sets for simple geodesics.