### 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)

switch (p_what) {
		case NOTIFICATION_ENTER_TREE: {
			get_tree()->connect("tree_changed", callable_mp(this, &SceneTreeEditor::_tree_changed));
			get_tree()->connect("tree_process_mode_changed", callable_mp(this, &SceneTreeEditor::_tree_process_mode_changed));
			get_tree()->connect("node_removed", callable_mp(this, &SceneTreeEditor::_node_removed));
			get_tree()->connect("node_renamed", callable_mp(this, &SceneTreeEditor::_node_renamed));
			get_tree()->connect("node_configuration_warning_changed", callable_mp(this, &SceneTreeEditor::_warning_changed));
			tree->connect("item_collapsed", callable_mp(this, &SceneTreeEditor::_cell_collapsed));
			_update_tree();
		} break;
		case NOTIFICATION_EXIT_TREE: {
			get_tree()->disconnect("tree_changed", callable_mp(this, &SceneTreeEditor::_tree_changed));
			get_tree()->disconnect("tree_process_mode_changed", callable_mp(this, &SceneTreeEditor::_tree_process_mode_changed));
			get_tree()->disconnect("node_removed", callable_mp(this, &SceneTreeEditor::_node_removed));
			get_tree()->disconnect("node_renamed", callable_mp(this, &SceneTreeEditor::_node_renamed));
			tree->disconnect("item_collapsed", callable_mp(this, &SceneTreeEditor::_cell_collapsed));
			get_tree()->disconnect("node_configuration_warning_changed", callable_mp(this, &SceneTreeEditor::_warning_changed));
		} break;
		case NOTIFICATION_THEME_CHANGED: {
			tree->add_theme_constant_override("icon_max_width", get_theme_constant(SNAME("class_icon_size"), SNAME("Editor")));
			_update_tree();
		} break;
	}
}
TreeItem *SceneTreeEditor::_find(TreeItem *p_node, const NodePath &p_path) {