### 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
project_menu->set_item_submenu(project_menu->get_item_index(VCS_MENU), "Version Control");
	add_editor_plugin(memnew(AudioBusesEditorPlugin(audio_bus_editor)));
	for (int i = 0; i < EditorPlugins::get_plugin_count(); i++) {
		add_editor_plugin(EditorPlugins::create(i));
	}
	for (const StringName &extension_class_name : GDExtensionEditorPlugins::get_extension_classes()) {
		add_extension_editor_plugin(extension_class_name);
	}
	GDExtensionEditorPlugins::editor_node_add_plugin = &EditorNode::add_extension_editor_plugin;
	GDExtensionEditorPlugins::editor_node_remove_plugin = &EditorNode::remove_extension_editor_plugin;
	for (int i = 0; i < plugin_init_callback_count; i++) {
		plugin_init_callbacks[i]();
	}
	resource_preview->add_preview_generator(Ref<EditorTexturePreviewPlugin>(memnew(EditorTexturePreviewPlugin)));
	resource_preview->add_preview_generator(Ref<EditorImagePreviewPlugin>(memnew(EditorImagePreviewPlugin)));
	resource_preview->add_preview_generator(Ref<EditorPackedScenePreviewPlugin>(memnew(EditorPackedScenePreviewPlugin)));
	resource_preview->add_preview_generator(Ref<EditorMaterialPreviewPlugin>(memnew(EditorMaterialPreviewPlugin)));
	resource_preview->add_preview_generator(Ref<EditorScriptPreviewPlugin>(memnew(EditorScriptPreviewPlugin)));
	resource_preview->add_preview_generator(Ref<EditorAudioStreamPreviewPlugin>(memnew(EditorAudioStreamPreviewPlugin)));
	resource_preview->add_preview_generator(Ref<EditorMeshPreviewPlugin>(memnew(EditorMeshPreviewPlugin)));
	resource_preview->add_preview_generator(Ref<EditorBitmapPreviewPlugin>(memnew(EditorBitmapPreviewPlugin)));
	resource_preview->add_preview_generator(Ref<EditorFontPreviewPlugin>(memnew(EditorFontPreviewPlugin)));
	resource_preview->add_preview_generator(Ref<EditorGradientPreviewPlugin>(memnew(EditorGradientPreviewPlugin)));
	{
		Ref<StandardMaterial3DConversionPlugin> spatial_mat_convert;
		spatial_mat_convert.instantiate();
		resource_conversion_plugins.push_back(spatial_mat_convert);
		Ref<ORMMaterial3DConversionPlugin> orm_mat_convert;
		orm_mat_convert.instantiate();
		resource_conversion_plugins.push_back(orm_mat_convert);