The current Linux shell uses GLFW, which significantly limits flexibility. For instance, there's no way to do a hybrid window (for instance, putting a GTK menu bar at the top of an otherwise-Flutter window).

See previous discussion in google/flutter-desktop-embedding#23

Given the lack of a standard toolkit, what exactly this should be is much less clear on Linux than on macOS or Windows. A likely outcome is that the shell would be implemented with GTK, but more research is needed.