The irreplaceability of the ice cores is why in my opinion, an independent backup refrigeration system (without common points of failure) is a necessity, along with an independent system that can isolate the primary chillers and activate the backup system should the primary chillers misbehave.

An emergency refrigeration system that works with consumable cryogenic material (say, dry ice or liquid nitrogen https://news.ycombinator.com/item?id=14127835) would be totally feasible and utterly inexpensive compared to the tragic costs of these cores melting.

I don't know the state of readiness of phase-change materials with appropriate transition temperatures (would need to be lower than -40 C) but if such a material exists, it'd work as well.