We do have a number and variety of toys and chews, including one like that. It's not entirely clear that the collecting-when-we-are-asleep behavior is entirely tied to just wanting something to chew on. She sleeps just fine through the night when closed in the front room, but loose she becomes very restless. I'm hoping that a crate in our bedroom will become her place and she'll settle in there, eventually not needing the door closed.
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