### Godot version

4.1.dev4

### System information

Windows 10 Godot 4.1.dev4

### Issue description

The watched property in the Multiplayer Synchronizer node is supposed to pass the variable through the network only when it changes, but as I use the network profiler, it is evidently not working as intended

### Steps to reproduce

Open 2 debug instances on the project I will give, use the network profiler and you will see the variable does not change, although it goes up to 780b/s on the network

### Minimal reproduction project

[minimalproject.zip](https://github.com/godotengine/godot/files/11659414/minimalproject.zip)

Ref<Texture2D> _get_class_icon(const Node *p_node);
	void _add_pressed();
	void _np_text_submitted(const String &p_newtext);
	void _tree_item_edited();
	void _tree_button_pressed(Object *p_item, int p_column, int p_id, MouseButton p_button);
	void _update_checked(const NodePath &p_prop, int p_column, bool p_checked);
	void _update_config();
	void _dialog_closed(bool p_confirmed);
	void _add_property(const NodePath &p_property, bool p_spawn = true, bool p_sync = true, bool p_watch = false);
	void _pick_node_filter_text_changed(const String &p_newtext);
	void _pick_node_select_recursive(TreeItem *p_item, const String &p_filter, Vector<Node *> &p_select_candidates);
	void _pick_node_filter_input(const Ref<InputEvent> &p_ie);
	void _pick_node_selected(NodePath p_path);
	void _pick_new_property();
	void _pick_node_property_selected(String p_name);
	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 _add_sync_property(String p_path);
protected:
	static void _bind_methods();
	void _notification(int p_what);
public:
	void edit(MultiplayerSynchronizer *p_object);
	MultiplayerSynchronizer *get_current() const { return current; }
	Button *get_pin() { return pin; }
	ReplicationEditor();
	~ReplicationEditor() {}
};
#endif