### 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
settings_menu->add_item(TTR("Manage Editor Features..."), SETTINGS_MANAGE_FEATURE_PROFILES);
#ifndef ANDROID_ENABLED
	settings_menu->add_item(TTR("Manage Export Templates..."), SETTINGS_MANAGE_EXPORT_TEMPLATES);
#endif
#if !defined(ANDROID_ENABLED) && !defined(WEB_ENABLED)
	settings_menu->add_item(TTR("Configure FBX Importer..."), SETTINGS_MANAGE_FBX_IMPORTER);
#endif
	help_menu = memnew(PopupMenu);
	help_menu->set_name(TTR("Help"));
	main_menu->add_child(help_menu);
	help_menu->connect("id_pressed", callable_mp(this, &EditorNode::_menu_option));
	ED_SHORTCUT_AND_COMMAND("editor/editor_help", TTR("Search Help"), Key::F1);
	ED_SHORTCUT_OVERRIDE("editor/editor_help", "macos", KeyModifierMask::ALT | Key::SPACE);
	help_menu->add_icon_shortcut(theme->get_icon(SNAME("HelpSearch"), EditorStringName(EditorIcons)), ED_GET_SHORTCUT("editor/editor_help"), HELP_SEARCH);
	help_menu->add_separator();
	help_menu->add_icon_shortcut(theme->get_icon(SNAME("ExternalLink"), EditorStringName(EditorIcons)), ED_SHORTCUT_AND_COMMAND("editor/online_docs", TTR("Online Documentation")), HELP_DOCS);
	help_menu->add_icon_shortcut(theme->get_icon(SNAME("ExternalLink"), EditorStringName(EditorIcons)), ED_SHORTCUT_AND_COMMAND("editor/q&a", TTR("Questions & Answers")), HELP_QA);
	help_menu->add_icon_shortcut(theme->get_icon(SNAME("ExternalLink"), EditorStringName(EditorIcons)), ED_SHORTCUT_AND_COMMAND("editor/community", TTR("Community")), HELP_COMMUNITY);
	help_menu->add_separator();
	help_menu->add_icon_shortcut(theme->get_icon(SNAME("ActionCopy"), EditorStringName(EditorIcons)), ED_SHORTCUT_AND_COMMAND("editor/copy_system_info", TTR("Copy System Info")), HELP_COPY_SYSTEM_INFO);
	help_menu->set_item_tooltip(-1, TTR("Copies the system info as a single-line text into the clipboard."));
	help_menu->add_icon_shortcut(theme->get_icon(SNAME("ExternalLink"), EditorStringName(EditorIcons)), ED_SHORTCUT_AND_COMMAND("editor/report_a_bug", TTR("Report a Bug")), HELP_REPORT_A_BUG);
	help_menu->add_icon_shortcut(theme->get_icon(SNAME("ExternalLink"), EditorStringName(EditorIcons)), ED_SHORTCUT_AND_COMMAND("editor/suggest_a_feature", TTR("Suggest a Feature")), HELP_SUGGEST_A_FEATURE);
	help_menu->add_icon_shortcut(theme->get_icon(SNAME("ExternalLink"), EditorStringName(EditorIcons)), ED_SHORTCUT_AND_COMMAND("editor/send_docs_feedback", TTR("Send Docs Feedback")), HELP_SEND_DOCS_FEEDBACK);
	help_menu->add_separator();
	if (!global_menu || !OS::get_singleton()->has_feature("macos")) {
		
		help_menu->add_icon_shortcut(theme->get_icon(SNAME("Godot"), EditorStringName(EditorIcons)), ED_SHORTCUT_AND_COMMAND("editor/about", TTR("About Godot")), HELP_ABOUT);
	}
	help_menu->add_icon_shortcut(theme->get_icon(SNAME("Heart"), EditorStringName(EditorIcons)), ED_SHORTCUT_AND_COMMAND("editor/support_development", TTR("Support Godot Development")), HELP_SUPPORT_GODOT_DEVELOPMENT);