### 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
left_menu_spacer->set_mouse_filter(Control::MOUSE_FILTER_PASS);
		title_bar->add_child(left_menu_spacer);
	}
	main_menu = memnew(MenuBar);
	title_bar->add_child(main_menu);
	main_menu->add_theme_style_override("hover", theme->get_stylebox(SNAME("MenuHover"), EditorStringName(EditorStyles)));
	main_menu->set_flat(true);
	main_menu->set_start_index(0); 
	main_menu->set_prefer_global_menu(global_menu);
	main_menu->set_switch_on_hover(true);
	file_menu = memnew(PopupMenu);
	file_menu->set_name(TTR("Scene"));
	main_menu->add_child(file_menu);
	main_menu->set_menu_tooltip(0, TTR("Operations with scene files."));
	accept = memnew(AcceptDialog);
	accept->set_unparent_when_invisible(true);
	save_accept = memnew(AcceptDialog);
	save_accept->set_unparent_when_invisible(true);
	save_accept->connect("confirmed", callable_mp(this, &EditorNode::_menu_option).bind((int)MenuOptions::FILE_SAVE_AS_SCENE));
	project_export = memnew(ProjectExportDialog);
	gui_base->add_child(project_export);
	dependency_error = memnew(DependencyErrorDialog);
	gui_base->add_child(dependency_error);
	dependency_fixer = memnew(DependencyEditor);
	gui_base->add_child(dependency_fixer);
	editor_settings_dialog = memnew(EditorSettingsDialog);
	gui_base->add_child(editor_settings_dialog);
	project_settings_editor = memnew(ProjectSettingsEditor(&editor_data));
	gui_base->add_child(project_settings_editor);
	scene_import_settings = memnew(SceneImportSettingsDialog);