Abate and Tovena's resonant Fuchsian-pole conjecture

Let $\nabla$ be a meromorphic connection on a Riemann surface $S$. Let $p_0$ be a Fuchsian pole, and let $\rho:=\operatorname{Res}_{p_0}\nabla$. Suppose

$$-1-\rho\in\mathbb{N}^*.$$

Abate and Tovena's conjecture. There is a neighborhood $U$ of $p_0$ such that every geodesic ray which enters into $U$ tends to $p_0$. The two rays of any geodesic which stays in $U$ tend to $p_0$. This conjecture extends the known local description around non-resonant Fuchsian poles to the resonant case, whose geodesic behavior is not established in the cited source.