Can I resolve the portion of this card's [All Turns] effect that follows "After" even if I don't fulfill the "by doing X" condition?
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No, you cannot.For this card's [All Turns] effect, if you do not place one [DS] trait Digimon card from your hand as this Digimon's bottom digivolution card, then you cannot resolve the portion that follows "After."
Source: EX12-028 Gusokumon