The chests in the Dungeon require specific gamestates, and the relevant characters in the active party. The first one you can see requires Sarah to be Dollified (and thus having Dollified Sarah in the active party when you walk up to it), the second requires a choice of either making Cassandra into a willing slave or becoming her sub (and thus requires having Cassandra in your active party when you walk up to it), the third requires Ashley to be cured of her "missing her soul disease" (and thus requires Cured Ashley to be in the active party when you walk up to it), and the fourth requires Kate (Rescue Girl) to simply be in the party (no specific gamestate beyond that, though the scenes tied to the thing you get out of the chest will require Kate to be in her Normal Personality, if she was recruited via taking control of her mind).