Imagine a number of chords of a sphere $S$ which nearly, but not quite, pass through
the center of $S$, in such a way that no pair of chords intersect:

I would like to connect pairs of chord endpoints by noncrossing paths on the surface of $S$ so
that the paths + the chords form a cycle. I can prove (for example, by induction) that
this is always possible if I am permitted
to use *arbitrary* paths on the surface. But what I would really like
to achieve is connecting
the chord endpoints by noncrossing *arcs of great circles*.

Henceforth consider the chords to pass exactly through the center of $S$, but pretend they do not intersect there. An example is shown left below, where three axes-chords are connected into a cycle. The example right below of coplanar chords shows that what I want to achieve is not always possible.

Q. Under what conditions can a set of chords through the center of $S$ be connected into a single cycle by noncrossing arcs of great circles? In particular, can this be achieved if no three chord endpoints lie on a great circle, i.e., are in general position in this sense?

There is a considerable literature on noncrossing geometric matchings in the plane, but I don't see that it applies to my question. Any pointers, ideas, or counterexamples welcomed!

(This arose in an investigation related (nonobviously) to an earlier question, "Untangling entwined rigid chains in 3-space".)