Relevant Information: Cross-Save player, migrated from PS4 and Battle.net. Forsaken license is tied to PS4, Shadowkeep license is tied to PC. Completed first step (defeat Ascendant Primeval) on PC. Quest disappeared at the same time I got confirmation of The Seething Heart step advancement at end of match/primeval defeat. Drifter did not advance quest on PC or PS4. Quest not available from Drifter / no abandoned quests.
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Hey there. Since you don't own Forsaken on PC you need to get the Seething Heart and complete the entire quest on PS4 by defeating the Ascendant Primeval.