### 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<InputEventKey> k = p_ie;
	if (k.is_valid()) {
		switch (k->get_keycode()) {
			case Key::UP:
			case Key::DOWN:
			case Key::PAGEUP:
			case Key::PAGEDOWN: {
				pick_node->get_scene_tree()->get_scene_tree()->gui_input(k);
				pick_node->get_filter_line_edit()->accept_event();
			} break;
			default:
				break;
		}
	}
}
void ReplicationEditor::_pick_node_selected(NodePath p_path) {