Why did you limit the Xenology Quest to one character per week? It doesn't make any sense if you wanna play on all of your characters..
Not only that, no, the quest disappears if you try to abandon it on one character to pick it up on another..
I've tried to resume the quest in any possible manner. IT DOESN'T WORK!
please, just make it available to ALL characters, once bought on one.
THANK YOU.
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Have you tried picking it up from Xur on each character? That's what I do.