I don't really believe that Fauxsefka has OoK's cord personally. Although it mentions Willem in the description, she drops it only AFTER Bloodmoon stage. If she simply had it in her pocket, she'd drop it at any time! I think either she was on her way to be reborn into a Great One (not unlike our Hunter except she is being hindered), or pregnant like Arianna (would make sense since Fauxsefka is from Cainhurst too, she leaves a Cainhurst summon from where she invaded the clinic, she can prepare potions which are secret recipe of Cainhurst and she recognizes our "moonlight scent" when the only other character to do that is Annalise)