### Tested versions

v4.2.stable.official [46dc27791]

### System information

Windows 11- Godot v4.2.stable.official [46dc27791] - Vulkan (Forward+)

### Issue description

When I switch picker mode the next time I open it should be the pick mode I selected last time but it's not

https://github.com/godotengine/godot/assets/46812028/522c8b55-14cf-4781-963f-a465a03c9432



### Steps to reproduce

Toggle a picker mode off and on again

### Minimal reproduction project (MRP)

N/A
root_name = EditorNode::adjust_scene_name_casing(root_name);
				file->set_current_path(root_name + "." + extensions.front()->get().to_lower());
			}
			file->popup_file_dialog();
			file->set_title(TTR("Save Scene As..."));
		} break;
		case FILE_SAVE_ALL_SCENES: {
			_save_all_scenes();
		} break;
		case FILE_RUN_SCENE: {
			project_run_bar->play_current_scene();
		} break;
		case FILE_EXPORT_PROJECT: {
			project_export->popup_export();
		} break;
		case FILE_EXTERNAL_OPEN_SCENE: {
			if (unsaved_cache && !p_confirmed) {
				confirmation->set_ok_button_text(TTR("Open"));
				confirmation->set_text(TTR("Current scene not saved. Open anyway?"));
				confirmation->popup_centered();
				break;
			}
			bool oprev = opening_prev;
			Error err = load_scene(external_file);
			if (err == OK && oprev) {
				previous_scenes.pop_back();
				opening_prev = false;
			}
		} break;
		case EDIT_UNDO: {