### 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
for (int i = 0; i < p_node->get_child_count(); i++) {
		Node *child = p_node->get_child(i);
		if (child->get_scene_file_path() == p_filename) {
			return true;
		}
		if (_validate_scene_recursive(p_filename, child)) {
			return true;
		}
	}
	return false;
}
int EditorNode::_save_external_resources() {
	
	int flg = 0;
	if (EDITOR_GET("filesystem/on_save/compress_binary_resources")) {
		flg |= ResourceSaver::FLAG_COMPRESS;
	}
	flg |= ResourceSaver::FLAG_REPLACE_SUBRESOURCE_PATHS;
	HashSet<String> edited_resources;
	int saved = 0;
	List<Ref<Resource>> cached;
	ResourceCache::get_cached_resources(&cached);
	for (Ref<Resource> res : cached) {
		if (!res->is_edited()) {