Existence of geodesic spirals for the Kobayashi--Fuks metric on planar domains

In this note, we discuss the following problem: Given a smoothly bounded strongly pseudoconvex domain $D$ in ${\mathbb{C}}^n$, can we guarantee the existence of geodesics for the Kobayashi–Fuks metric which ``spiral around" in the interior of $D$? We find an affirmative answer to the above question for $n=1$ when $D$ is not simply connected.