### 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)

void _update_selection(TreeItem *item);
	void _node_script_changed(Node *p_node);
	void _node_visibility_changed(Node *p_node);
	void _update_visibility_color(Node *p_node, TreeItem *p_item);
	void _selection_changed();
	Node *get_scene_node();
	Variant get_drag_data_fw(const Point2 &p_point, Control *p_from);
	bool can_drop_data_fw(const Point2 &p_point, const Variant &p_data, Control *p_from) const;
	void drop_data_fw(const Point2 &p_point, const Variant &p_data, Control *p_from);
	void _empty_clicked(const Vector2 &p_pos, MouseButton p_button);
	void _rmb_select(const Vector2 &p_pos, MouseButton p_button = MouseButton::RIGHT);
	void _warning_changed(Node *p_for_node);
	Timer *update_timer = nullptr;
	List<StringName> *script_types;
	bool _is_script_type(const StringName &p_type) const;
	Vector<StringName> valid_types;
public:
	
	void _update_tree(bool p_scroll_to_selected = false);
	void set_filter(const String &p_filter);
	String get_filter() const;
	String get_filter_term_warning();
	void set_show_all_nodes(bool p_show_all_nodes);
	void set_as_scene_tree_dock();
	void set_display_foreign_nodes(bool p_display);
	void set_marked(const HashSet<Node *> &p_marked, bool p_selectable = false, bool p_children_selectable = true);
	void set_marked(Node *p_marked, bool p_selectable = false, bool p_children_selectable = true);
	void set_selected(Node *p_node, bool p_emit_selected = true);
	Node *get_selected();
	void set_can_rename(bool p_can_rename) { can_rename = p_can_rename; }