### 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)

String adding_prop_path = String(adding_node_path) + ":" + p_name;
	_add_sync_property(adding_prop_path);
}
ReplicationEditor::ReplicationEditor() {
	set_v_size_flags(SIZE_EXPAND_FILL);
	set_custom_minimum_size(Size2(0, 200) * EDSCALE);
	delete_dialog = memnew(ConfirmationDialog);
	delete_dialog->connect("canceled", callable_mp(this, &ReplicationEditor::_dialog_closed).bind(false));
	delete_dialog->connect("confirmed", callable_mp(this, &ReplicationEditor::_dialog_closed).bind(true));
	add_child(delete_dialog);
	error_dialog = memnew(AcceptDialog);
	error_dialog->set_ok_button_text(TTR("Close"));
	error_dialog->set_title(TTR("Error!"));
	add_child(error_dialog);
	VBoxContainer *vb = memnew(VBoxContainer);
	vb->set_v_size_flags(SIZE_EXPAND_FILL);
	add_child(vb);
	pick_node = memnew(SceneTreeDialog);
	add_child(pick_node);
	pick_node->register_text_enter(pick_node->get_filter_line_edit());
	pick_node->set_title(TTR("Pick a node to synchronize:"));
	pick_node->connect("selected", callable_mp(this, &ReplicationEditor::_pick_node_selected));
	pick_node->get_filter_line_edit()->connect("text_changed", callable_mp(this, &ReplicationEditor::_pick_node_filter_text_changed));
	pick_node->get_filter_line_edit()->connect("gui_input", callable_mp(this, &ReplicationEditor::_pick_node_filter_input));
	prop_selector = memnew(PropertySelector);
	add_child(prop_selector);