The only suggestion of independent power supply that has emerged was after SR111 and that was the voice recorder.
Airbus CVR designs currently record 4-channels - the Capt, F/O and 3rd occupant crew mic signals, and the CVR area mic. Since the crew microphones within modern aircraft are all 'powered' (electret), these will be lost if electrical power is lost. The dedicated CVR microphone could be powered from an internal supply sourced from a battery supply, so that would remain active.