  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.
