### Godot version

v4.2.dev.custom_build [0ca854232]

### System information

Pop Os 22.04 lts 12th Gen Intel® Core™ i7-12700H × 20 NVIDIA RTX A2000 8GB 32.0GB  ram

### Issue description

A node that has a script attached to it and it has a couple of child nodes (type of a child node doesn't matter). In the script you have an export variable of type Node. You save the scene into a file and you add it in level scene. When you start to duplicate that scene in the level and you'll hover over exported variable to show a node path, it will show an incorrect path. 


https://github.com/godotengine/godot/assets/33091666/75088d1b-c014-41e0-bae3-46610eccbad1



### Steps to reproduce

1. create a scene with a node and a child node.
2. attach a script to the parent node and create an export variable of type Node
3. assign child node to that exported variable through the editor
4. save the scene and create a level scene where you are going to add just saved scene.
5. _Possibly requires a reboot of the engine before seeing the bug_
6. start duplicating nodes in the level scene

### Minimal reproduction project

[NodeTest.zip](https://github.com/godotengine/godot/files/12781547/NodeTest.zip)

int enabled_notification = 0;
	if (prev_enabled && !next_enabled) {
		enabled_notification = NOTIFICATION_DISABLED;
	} else if (!prev_enabled && next_enabled) {
		enabled_notification = NOTIFICATION_ENABLED;
	}
	_propagate_process_owner(data.process_owner, pause_notification, enabled_notification);
#ifdef TOOLS_ENABLED
	
	
	if (Engine::get_singleton()->is_editor_hint()) {
		get_tree()->emit_signal(SNAME("tree_process_mode_changed"));
	}
#endif
}
void Node::_propagate_pause_notification(bool p_enable) {
	bool prev_can_process = _can_process(!p_enable);
	bool next_can_process = _can_process(p_enable);
	if (prev_can_process && !next_can_process) {
		notification(NOTIFICATION_PAUSED);
	} else if (!prev_can_process && next_can_process) {
		notification(NOTIFICATION_UNPAUSED);
	}
	data.blocked++;
	for (KeyValue<StringName, Node *> &K : data.children) {
		K.value->_propagate_pause_notification(p_enable);
	}
	data.blocked--;
}
Node::ProcessMode Node::get_process_mode() const {