Can superconducting electron pairs be annihilated when the pairs flow in an eternal supercurrent?

[StanislavDolgopolov]

A simple question in conventional theories of superconductivity seems to be open. Imagine, in a mercury ring (superconductivity below T_c=4.15 K) we establish a persistent supercurrent. Then we organize temperature cycles (T-cycles) in the cryostat, say from 3 K to 2.5 K and back. According to the BCS theory of superconductivity, the pair density decreases at warming, i.e. a not negligible fraction of pairs annihilates; the same fraction of pairs emerges back at cooling. Annihilated pairs lose their ordered supercurrent momentum on the atom lattice, so the BCS-supercurrent must decrease at warming; newly created pairs do not experience any electromotive-force (EMF), since the EMF is no longer available in the ring. Hence, according to the BCS theory, the supercurrent must decrease at every T-cycle and dissipate after a number of T-cycles. However, in all experiments the supercurrent remains constant and, thus, the pair recombination (assumed in BCS) doesn’t take place (note, every cryostat device produces not negligible temperature fluctuations, so every observation of long-lived supercurrents is the experiment with T-cycles).
Do the pairs really annihilate in the eternal supercurrent?
So far this paradox is open.