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You can manage & track your card collection and decks once you create an account and login.

What does ‘each battle’ entail regarding green icons? For example, if my opponent has 3 attacks on my one character, every time I flip a green battle icon for each individual battle that turn, I can add one green icon for each battle, totaling 3?

Yes. You can get one green battle icon per battle.
Source: Wizards of the Coast (2019-03-15)    【Link to Ruling】
Tags: Battle Icons