The term is “flux pinning”, and it only applies to the “quantum lock” effect. That is the specifically static hovering effect.
The diamagnetism, importantly this means repulsion of both poles simultaneously and equally (this is how you can have these magnets spin, a regular magnet repels same poles and attracts opposites, diamagnets repel both poles), is simply a characteristic of the superconductor, but it alone would just repel the object off.
Here is a timestamped link to NileRed’s YBCO video that visually describes the flux pinning:
And here’s a timestamped link to Ben Krasnow’s Applied Science YBCO video where he shows a close up of the crystal’s cross section that shows the imperfections that allow the magnetic field through for the pinning effect:
Flux pinning of a superconductor should be able to hold it steady below a magnet, and the magnet should be able to drag the superconductor with it when moving (within reasonable weight limits of course). These will demonstrate for sure that levitation is not simply a force equilibrium between gravity, magnetic repulsion and one corner of the material resting on the surface.
There is a video from the Korean team showing LK99 moving when both poles of a large magnet is swung nearby, however the effect was a bit weak to conclusive.
If we develop methods of creating these superconductors with perfect crystal composition then there will only be the repulsion, allowing for levitation in a bowl shaped superconductor, but this “hanging levitation” would be impossible.
Perhaps we will develop manufacturing techniques to induce specific imperfections into the material to ensure predictable flux pinning; it seems like a useful, and wildly interesting side effect.
The best thing about the NileRed superconductor video is it shows him initially failing to reproduce a YBCO superconductor after having already succeeded once before!
It goes to show how difficult manufacture, or in the case of the LK-99 news cycle “reproduction”, of these materials really is, and YBCO was a well documented area of superconductor manufacture.
But I am surprised and very impressed with how the researchers choose to use it, and that bug eye shaped lens array is pure scifi fun!
The demo in the video, timestamped below, showing why they used such a design to create a single opening behind the array is amazing in its own right.
https://m.youtube.com/watch?v=O15wUC5D7zE&t=85