### Godot version

4.1???- 4.1.beta1 - 4.1.beta3 - 030c1a950e8cee19e72104b4fb79caba17a2d678

### System information

Tested only Linux versions.

### Issue description

**Regression from 4.0.3.stable:**
In the editor UI set the  `Node->process_mode` to `Disabled` or `When Paused`-> in the scene editor the subtree is grayed out. Save the scene and close the editor. 
It doesn*t need to be the current scene. 
Start it again, the subtree in the scene is no longer gray. The *.tscn file is updated correctly, e.g.:
```
  [node name="Node2D" type="Node2D"]
+process_mode = 4
```
 Workaround is to toggle the process mode again,
 close and open or reload the scene,
(or disable the `interface/scene_tabs/restore_scenes_on_load` option ...)

### Image gallery
![image](https://github.com/godotengine/godot/assets/4047289/b6a90b34-c9ec-4380-b953-f1e64cf72934)
![image](https://github.com/godotengine/godot/assets/4047289/978d4ab1-553b-49e6-bb76-6b18c22a942b)
![image](https://github.com/godotengine/godot/assets/4047289/4254f160-75d5-4186-9802-53b487d8d2f3)

if (!p_node) {
		return;
	}
	
	
	bool part_of_subscene = false;
	if (!display_foreign && p_node->get_owner() != get_scene_node() && p_node != get_scene_node()) {
		if ((show_enabled_subscene || can_open_instance) && p_node->get_owner() && (get_scene_node()->is_editable_instance(p_node->get_owner()))) {
			part_of_subscene = true;
			
		} else {
			return;
		}
	} else {
		part_of_subscene = p_node != get_scene_node() && get_scene_node()->get_scene_inherited_state().is_valid() && get_scene_node()->get_scene_inherited_state()->find_node_by_path(get_scene_node()->get_path_to(p_node)) >= 0;
	}
	TreeItem *item = tree->create_item(p_parent);
	item->set_text(0, p_node->get_name());
	if (can_rename && !part_of_subscene) {
		item->set_editable(0, true);
	}
	item->set_selectable(0, true);
	if (can_rename) {
		bool collapsed = p_node->is_displayed_folded();
		if (collapsed) {
			item->set_collapsed(true);
		}