MM actually I think it increases the number of attacks with that weapon only, no matter the hand.
Thus if you had a katana +3 in your fighting hand, and Balm in your off hand, and you normally get 4 attacks (3 with primary, 1 with balm), you would get 5 attacks (3 with primary, 2 with balm).
SO it doesn't speed up your primary weapon, but I believe you get an extra attack with Balm. There are a couple weapons like this.
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