### 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
gui_base->add_child(scene_import_settings);
	audio_stream_import_settings = memnew(AudioStreamImportSettingsDialog);
	gui_base->add_child(audio_stream_import_settings);
	fontdata_import_settings = memnew(DynamicFontImportSettingsDialog);
	gui_base->add_child(fontdata_import_settings);
	export_template_manager = memnew(ExportTemplateManager);
	gui_base->add_child(export_template_manager);
	feature_profile_manager = memnew(EditorFeatureProfileManager);
	gui_base->add_child(feature_profile_manager);
	build_profile_manager = memnew(EditorBuildProfileManager);
	gui_base->add_child(build_profile_manager);
	about = memnew(EditorAbout);
	gui_base->add_child(about);
	feature_profile_manager->connect("current_feature_profile_changed", callable_mp(this, &EditorNode::_feature_profile_changed));
#if !defined(ANDROID_ENABLED) && !defined(WEB_ENABLED)
	fbx_importer_manager = memnew(FBXImporterManager);
	gui_base->add_child(fbx_importer_manager);
#endif
	warning = memnew(AcceptDialog);
	warning->set_unparent_when_invisible(true);
	warning->add_button(TTR("Copy Text"), true, "copy");
	warning->connect("custom_action", callable_mp(this, &EditorNode::_copy_warning));
	ED_SHORTCUT("editor/next_tab", TTR("Next Scene Tab"), KeyModifierMask::CMD_OR_CTRL + Key::TAB);
	ED_SHORTCUT("editor/prev_tab", TTR("Previous Scene Tab"), KeyModifierMask::CMD_OR_CTRL + KeyModifierMask::SHIFT + Key::TAB);
	ED_SHORTCUT("editor/filter_files", TTR("Focus FileSystem Filter"), KeyModifierMask::CMD_OR_CTRL + KeyModifierMask::ALT + Key::P);
	command_palette = EditorCommandPalette::get_singleton();
	command_palette->set_title(TTR("Command Palette"));
	gui_base->add_child(command_palette);
	file_menu->add_shortcut(ED_SHORTCUT_AND_COMMAND("editor/new_scene", TTR("New Scene"), KeyModifierMask::CMD_OR_CTRL + Key::N), FILE_NEW_SCENE);
	file_menu->add_shortcut(ED_SHORTCUT_AND_COMMAND("editor/new_inherited_scene", TTR("New Inherited Scene..."), KeyModifierMask::CMD_OR_CTRL + KeyModifierMask::SHIFT + Key::N), FILE_NEW_INHERITED_SCENE);