The ranges for the pool of four augments for each piece seem to be something like:
A <= B <= C <= D
HQ1: A to C
HQ2: A to D
HQ3: B to D
So the HQ1 and HQ2 have the same minimum and the HQ2 and HQ3 have the same maximum values. I'm pretty sure you can get the maximum value on any augment from the pool of four augments for all pieces on either HQ2 or HQ3, aside from ones which have much larger ranges like HP or MP.
For the HQ guaranteed augment, it's different, HQ1 and HQ2 seem to generally have the same range, with HQ3's range (minimum and maximum) being higher.
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