### 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)

if (!p_item) {
		return;
	}
	NodePath np = p_item->get_metadata(0);
	Node *node = get_node(np);
	if (!p_filter.is_empty() && ((String)node->get_name()).findn(p_filter) != -1) {
		p_select_candidates.push_back(node);
	}
	TreeItem *c = p_item->get_first_child();
	while (c) {
		_pick_node_select_recursive(c, p_filter, p_select_candidates);
		c = c->get_next();
	}
}
void ReplicationEditor::_pick_node_filter_input(const Ref<InputEvent> &p_ie) {