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.