### 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 _test_update_tree();
	bool _update_filter(TreeItem *p_parent = nullptr, bool p_scroll_to_selected = false);
	bool _item_matches_all_terms(TreeItem *p_item, PackedStringArray p_terms);
	void _tree_changed();
	void _tree_process_mode_changed();
	void _node_removed(Node *p_node);
	void _node_renamed(Node *p_node);
	TreeItem *_find(TreeItem *p_node, const NodePath &p_path);
	void _notification(int p_what);
	void _selected_changed();
	void _deselect_items();
	void _rename_node(Node *p_node, const String &p_name);
	void _cell_collapsed(Object *p_obj);
	uint64_t last_hash;
	bool can_rename;
	bool can_open_instance;
	bool updating_tree = false;
	bool show_enabled_subscene = false;
	bool is_scene_tree_dock = false;
	void _renamed();
	HashSet<Node *> marked;
	bool marked_selectable = false;
	bool marked_children_selectable = false;
	bool display_foreign = false;
	bool tree_dirty = true;
	bool pending_test_update = false;
	static void _bind_methods();
	void _cell_button_pressed(Object *p_item, int p_column, int p_id, MouseButton p_button);
	void _toggle_visible(Node *p_node);
	void _cell_multi_selected(Object *p_object, int p_cell, bool p_selected);