I ended up with Abby's (OK, Caroline's) Ornate Necklace in a multiplayer game on Switch even though Abby acknowledged that I'd found it.
Problem was: I'd found it but put it in a Chest on the Farm. And then bumped into Abby to get the Necklace dialog, after which the quest as completed. Went back to the Chest and sure enough the Necklace is still there! If you still have it in your inventory and the quest is done, just pop it into a Chest and leave it there.
It's mine now, I wear it, and look smashing in it. A nice gift from my favorite mobile mineral muncher in the game. Caroline's always trying to ask me about it during the Flower Dance but ya know, I just gotta circulate. You know, follow Haley around to get the Dance steps down, flick rubber bands at the Wizard creepin' through the trees, slip Willy a Fishing Rod when Lewis isn't watching, compliment Elliot on his Shirt. Yeah, just no time for heavy provenance discussions...