That is not what the rules say:
Exception: Some Reward cards include the text “keep this card.” In these cases, one participating character keeps the Reward card until the card directs that it should be discarded.
Discarded Reward cards are returned to their Stage decks.
Discarded rewards are returned to the stage deck. The cards are shuffled before you start drawing them (see Resolving a Dungeon Stage, page 10)