### 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)

class MultiplayerSynchronizer;
class AcceptDialog;
class LineEdit;
class Tree;
class TreeItem;
class PropertySelector;
class SceneTreeDialog;
class ReplicationEditor : public VBoxContainer {
	GDCLASS(ReplicationEditor, VBoxContainer);
private:
	MultiplayerSynchronizer *current = nullptr;
	AcceptDialog *error_dialog = nullptr;
	ConfirmationDialog *delete_dialog = nullptr;
	Button *add_pick_button = nullptr;
	Button *add_from_path_button = nullptr;
	LineEdit *np_line_edit = nullptr;
	Label *drop_label = nullptr;
	Ref<SceneReplicationConfig> config;
	NodePath deleting;
	Tree *tree = nullptr;
	PropertySelector *prop_selector = nullptr;
	SceneTreeDialog *pick_node = nullptr;
	NodePath adding_node_path;
	Button *pin = nullptr;