Unfortunately, developers Arrowhead have no plans to include a story mode in Helldivers 2, as revealed by a comment from new CEO Shams Jorjani. He confirmed that a cooperative story mode with epic and tragic moments will not be included in the game, explaining it would require the creation of an entirely new game.
However, the studio is planning to go on a well-deserved summer vacation, which may slightly affect the current processes, but after the vacation they will try to stick to the schedule of updates and patch releases as before.