Vinegar is supposed to remove alkaline battery corrosion. I used a q-tip and vinegar to restore my Bose QC2s, and then was careful to never store them with the battery inside. Unfortunately, despite this, they developed the same problem again (not turning on when a good battery is inserted), and this time vinegar has not been able to make them work.
I can barely see inside, but the anode at the bottom ("-" side) looks corroded brown rather than green. Perhaps a different issue. I can't believe how fragile these super expensive headphones have been. I'll try more drastic measures to clean inside there...not much to lose at this point.