This seems like a great question to me and I’m bummed I can’t answer it. But here’s a toy model that might help a bit.
Take a 2-dimensional spacetime shaped like the surface of a vertical cylinder, with space being the 1-dimensional equatorial circles, and time going vertically. Some of the straight lines in this space are slanted lines just going around and around the cylinder forever, and objects following those as world lines would sort of appear to oscillate around a point traveling along an exact vertical world line.
Anyway that model’s only 2-dimensional, and the bigger problem is it’s not the right type of geometry (it’s Riemannian not Lorentzian). Also the cylinder is flat, not curved. But maybe it still helps.
Really neat paper, thank you!