### 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)

selected = nullptr;
	can_rename = p_can_rename;
	can_open_instance = p_can_open_instance;
	editor_selection = nullptr;
	if (p_label) {
		Label *label = memnew(Label);
		label->set_theme_type_variation("HeaderSmall");
		label->set_position(Point2(10, 0));
		label->set_text(TTR("Scene Tree (Nodes):"));
		add_child(label);
	}
	tree = memnew(Tree);
	tree->set_anchor(SIDE_RIGHT, ANCHOR_END);
	tree->set_anchor(SIDE_BOTTOM, ANCHOR_END);
	tree->set_begin(Point2(0, p_label ? 18 : 0));
	tree->set_end(Point2(0, 0));
	tree->set_allow_reselect(true);
	tree->add_theme_constant_override("button_margin", 0);
	add_child(tree);
	SET_DRAG_FORWARDING_GCD(tree, SceneTreeEditor);
	if (p_can_rename) {
		tree->set_allow_rmb_select(true);
		tree->connect("item_mouse_selected", callable_mp(this, &SceneTreeEditor::_rmb_select));
		tree->connect("empty_clicked", callable_mp(this, &SceneTreeEditor::_empty_clicked));
	}
	tree->connect("cell_selected", callable_mp(this, &SceneTreeEditor::_selected_changed));
	tree->connect("item_edited", callable_mp(this, &SceneTreeEditor::_renamed));