### 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
TTR("The current scene has unsaved changes.\nReload the saved scene anyway? This action cannot be undone."));
				confirmation->popup_centered();
				break;
			}
			int cur_idx = editor_data.get_edited_scene();
			_remove_edited_scene();
			Error err = load_scene(filename);
			if (err != OK) {
				ERR_PRINT("Failed to load scene");
			}
			editor_data.move_edited_scene_to_index(cur_idx);
			EditorUndoRedoManager::get_singleton()->clear_history(false, editor_data.get_current_edited_scene_history_id());
			scene_tabs->set_current_tab(cur_idx);
		} break;
		case FILE_SHOW_IN_FILESYSTEM: {
			String path = editor_data.get_scene_path(editor_data.get_edited_scene());
			if (!path.is_empty()) {
				FileSystemDock::get_singleton()->navigate_to_path(path);
			}
		} break;
		case RUN_SETTINGS: {
			project_settings_editor->popup_project_settings();
		} break;
		case FILE_INSTALL_ANDROID_SOURCE: {
			if (p_confirmed) {
				export_template_manager->install_android_template();
			} else {
				if (DirAccess::exists("res:android/build")) {
					remove_android_build_template->popup_centered();
				} else if (export_template_manager->can_install_android_template()) {