The heat is certainly part of the energy storage in lightsail, if the heat generated by the compression wasn't being stored in water and then fed back in during

The heat is certainly part of the energy storage in lightsail, if the heat generated by the compression wasn't being stored in water and then fed back in during expansion then that energy would be lost. Is just that with lightsail it is a minor part of the storage, the majority being pressure. They are two very different solutions that approach two similar but differently structured thermodynamic problems, lightsail's system being high pressure/low temperature, and these folk doing a low pressure/high temperature store.