### 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
file_menu->add_separator();
		file_menu->add_shortcut(ED_SHORTCUT_AND_COMMAND("editor/file_quit", TTR("Quit"), KeyModifierMask::CMD_OR_CTRL + Key::Q), FILE_QUIT, true);
	}
	project_menu = memnew(PopupMenu);
	project_menu->set_name(TTR("Project"));
	main_menu->add_child(project_menu);
	project_menu->add_shortcut(ED_SHORTCUT_AND_COMMAND("editor/project_settings", TTR("Project Settings..."), Key::NONE, TTR("Project Settings")), RUN_SETTINGS);
	project_menu->connect("id_pressed", callable_mp(this, &EditorNode::_menu_option));
	project_menu->add_separator();
	project_menu->add_item(TTR("Version Control"), VCS_MENU);
	project_menu->add_separator();
	project_menu->add_shortcut(ED_SHORTCUT_AND_COMMAND("editor/export", TTR("Export..."), Key::NONE, TTR("Export")), FILE_EXPORT_PROJECT);
#ifndef ANDROID_ENABLED
	project_menu->add_item(TTR("Install Android Build Template..."), FILE_INSTALL_ANDROID_SOURCE);
	project_menu->add_item(TTR("Open User Data Folder"), RUN_USER_DATA_FOLDER);
#endif
	project_menu->add_separator();
	project_menu->add_item(TTR("Customize Engine Build Configuration..."), TOOLS_BUILD_PROFILE_MANAGER);
	project_menu->add_separator();
	tool_menu = memnew(PopupMenu);
	tool_menu->set_name("Tools");
	tool_menu->connect("index_pressed", callable_mp(this, &EditorNode::_tool_menu_option));
	project_menu->add_child(tool_menu);
	project_menu->add_submenu_item(TTR("Tools"), "Tools");
	tool_menu->add_item(TTR("Orphan Resource Explorer..."), TOOLS_ORPHAN_RESOURCES);
	tool_menu->add_item(TTR("Upgrade Mesh Surfaces..."), TOOLS_SURFACE_UPGRADE);
	project_menu->add_separator();
	project_menu->add_shortcut(ED_SHORTCUT("editor/reload_current_project", TTR("Reload Current Project")), RELOAD_CURRENT_PROJECT);
	ED_SHORTCUT_AND_COMMAND("editor/quit_to_project_list", TTR("Quit to Project List"), KeyModifierMask::CTRL + KeyModifierMask::SHIFT + Key::Q);