### 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)

#include "replication_editor.h"
#include "../multiplayer_synchronizer.h"
#include "editor/editor_node.h"
#include "editor/editor_scale.h"
#include "editor/editor_settings.h"
#include "editor/editor_undo_redo_manager.h"
#include "editor/gui/scene_tree_editor.h"
#include "editor/inspector_dock.h"
#include "editor/property_selector.h"
#include "scene/gui/dialogs.h"
#include "scene/gui/separator.h"
#include "scene/gui/tree.h"
void ReplicationEditor::_pick_node_filter_text_changed(const String &p_newtext) {