When Brain Ghost Dirk disappeared as a result from Jake dying and losing Aranea's power, why did his sword stay behind?

Dirk is a Living Being, while the Sword is just an object. It seems similar to how Roxy can easily make Cubes but struggles with more complex things. In the same way, a Katana is something simple and streamlined enough that you don’t need to keep believing it’s there for it to be there, Dirk meanwhile is a projection of Dirk’s personality made to protect Jake, that’s a bit harder to maintain than a static object.