### 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)

TreeItem *root_item = pick_node->get_scene_tree()->get_scene_tree()->get_root();
	Vector<Node *> select_candidates;
	Node *to_select = nullptr;
	String filter = pick_node->get_filter_line_edit()->get_text();
	_pick_node_select_recursive(root_item, filter, select_candidates);
	if (!select_candidates.is_empty()) {
		for (int i = 0; i < select_candidates.size(); ++i) {
			Node *candidate = select_candidates[i];
			if (((String)candidate->get_name()).to_lower().begins_with(filter.to_lower())) {
				to_select = candidate;
				break;
			}
		}
		if (!to_select) {
			to_select = select_candidates[0];
		}
	}
	pick_node->get_scene_tree()->set_selected(to_select);
}
void ReplicationEditor::_pick_node_select_recursive(TreeItem *p_item, const String &p_filter, Vector<Node *> &p_select_candidates) {