That behaviour looks pretty normal for a Type-I superconductor, the "locking on" you are referencing is a property of Type-2 Superconductors. For more information search Meißner-Ochsenfeld Effect (ideal diamagnetism, type 1 SC) and flux pinning (type 2)

I have a fair bit of experience with superconductivity and the submitted manuscript on arXiv looks solid, I didnt notice anything suspicious

Edit: the falling down may be because the material is not superconductive throughout, it looks like it falls down, rotates because of the magnetic field (maybe the non superconductive phase is magnetic?) then pops back up because the magnetic field of the magnet is once again strong enough to lift the sample (the Meißner effect dispels the field inside the Superconducter by generating shielding currents just below the surface, thus "mirroring" the field of the permanent magnet)