### 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 SceneTreeDialog::_update_theme() {
	filter->set_right_icon(tree->get_theme_icon(SNAME("Search"), SNAME("EditorIcons")));
	for (TextureRect *trect : valid_type_icons) {
		trect->set_custom_minimum_size(Vector2(get_theme_constant(SNAME("class_icon_size"), SNAME("Editor")), 0));
		trect->set_texture(EditorNode::get_singleton()->get_class_icon(trect->get_meta("type")));
	}
}
void SceneTreeDialog::_notification(int p_what) {
	switch (p_what) {
		case NOTIFICATION_VISIBILITY_CHANGED: {
			if (is_visible()) {
				tree->update_tree();
				// S
				filter->call_deferred(SNAME("grab_focus"));
			}
		} break;
		case NOTIFICATION_ENTER_TREE: {
			connect("confirmed", callable_mp(this, &SceneTreeDialog::_select));
			_update_theme();
		} break;
		case NOTIFICATION_THEME_CHANGED: {
			_update_theme();
		} break;
		case NOTIFICATION_EXIT_TREE: {
			disconnect("confirmed", callable_mp(this, &SceneTreeDialog::_select));
		} break;
	}
}
void SceneTreeDialog::_cancel() {