### 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)

connect_to_script_mode = p_enable;
	update_tree();
}
void SceneTreeEditor::set_connecting_signal(bool p_enable) {
	connecting_signal = p_enable;
	update_tree();
}
void SceneTreeEditor::_bind_methods() {
	ClassDB::bind_method(D_METHOD("_update_tree"), &SceneTreeEditor::_update_tree, DEFVAL(false)); // Still used by UndoRedo.
	ClassDB::bind_method("_rename_node", &SceneTreeEditor::_rename_node);
	ClassDB::bind_method("_test_update_tree", &SceneTreeEditor::_test_update_tree);
	ClassDB::bind_method(D_METHOD("update_tree"), &SceneTreeEditor::update_tree);
	ADD_SIGNAL(MethodInfo("node_selected"));
	ADD_SIGNAL(MethodInfo("node_renamed"));
	ADD_SIGNAL(MethodInfo("node_prerename"));
	ADD_SIGNAL(MethodInfo("node_changed"));
	ADD_SIGNAL(MethodInfo("nodes_dragged"));
	ADD_SIGNAL(MethodInfo("nodes_rearranged", PropertyInfo(Variant::ARRAY, "paths"), PropertyInfo(Variant::NODE_PATH, "to_path"), PropertyInfo(Variant::INT, "type")));
	ADD_SIGNAL(MethodInfo("files_dropped", PropertyInfo(Variant::PACKED_STRING_ARRAY, "files"), PropertyInfo(Variant::NODE_PATH, "to_path"), PropertyInfo(Variant::INT, "type")));
	ADD_SIGNAL(MethodInfo("script_dropped", PropertyInfo(Variant::STRING, "file"), PropertyInfo(Variant::NODE_PATH, "to_path")));
	ADD_SIGNAL(MethodInfo("rmb_pressed", PropertyInfo(Variant::VECTOR2, "position")));
	ADD_SIGNAL(MethodInfo("open"));
	ADD_SIGNAL(MethodInfo("open_script"));
}
SceneTreeEditor::SceneTreeEditor(bool p_label, bool p_can_rename, bool p_can_open_instance) {