Tested, and Unaware does take effect.
However, it does not make Stored Power completely powerless. Stored Power still increases in power from the stat boosts because of its effect, yet does not get the Special Attack increase itself to draw upon to increase the damage done.
With Quagsire, there was found to be a 400% difference in the damage done by Stored Power vs a Water Absorb Quagsire over an Unaware Quagsire, with the Unaware Quagsire taking much less damage. However, the Quagsire was still able to be KOd, something which could not have been done if Stored Power was still a 20 base power attack or whatever the hell it is.
