My theory is that Pearl is literally Rose's scabbard. A scabbard protects a sword from getting dull. It makes sure that the sword stays in an optimum condition. Pearl protects Rose and makes sure that her health stays in perfect condition.Pearl has said that,"nothing could fit together so perfectly" she was talking about herself and Rose. That's why she runs ways clutching the scabbard as if her life depended on it.

During the episode, I noticed something. When Steven was jumping on the ledges and Pearl turned and gave him a death stare. She thought to herself "I wish he would just fall" because she blamed him for Rose no longer being there.
Then Steven saw in her eyes that she did blame him and he fell. Pearl saw that Steven had noticed what she felt and wanted to explain but he had fell. She then walked over to the other side of the ledge and sat down crying. She thought that Rose was disappointed in her for blaming Steven. When Steven hugged her she felt as if both of them were forgiving her.