When my opponent's Digimon attacks and checks my security, I activate this card's effect to gain 1 memory, moving the memory counter to 1 on my side. Does the attack end, even if my opponent's Digimon has checks remaining from <Security Attack +1>?
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No, as long as there are attacks and effects that have triggered but haven't finished resolving yet, the attack continues regardless of the memory gauge's current position.
Source: BT4-097 Kari Kamiya