The results were very disappointing.
ONLY the shared cooldown of other batteries was reduced. The cooldown of the activated item (Battery - Energy Amplifier) remained the same at 1 minute. But the weapons battery shared cooldown went to 20 seconds instead of 30 seconds.
To be clear:
Activating Battery - Energy Amplifier results in a 1 min Energy Amplifier cooldown and a 30 second Weapons battery cooldown.
With Rare Blue Quartermaster:
Activating Battery - Energy Amplifier results in a 1 min Energy Amplifier cooldown and a 20 second Weapons battery cooldown.
This is not useful to me since I only carry the Battery - Energy Amplifier and was trying to reduce its 1 min cooldown.
An expensive experiment! I almost tried it with a green version first. It would have been a lot cheaper experiment!