### 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)

data.ready_notified = true;
	data.blocked++;
	for (KeyValue<StringName, Node *> &K : data.children) {
		K.value->_propagate_ready();
	}
	data.blocked--;
	notification(NOTIFICATION_POST_ENTER_TREE);
	if (data.ready_first) {
		data.ready_first = false;
		notification(NOTIFICATION_READY);
		emit_signal(SceneStringNames::get_singleton()->ready);
	}
}
void Node::_propagate_enter_tree() {
	
	if (data.parent) {
		data.tree = data.parent->data.tree;
		data.depth = data.parent->data.depth + 1;
	} else {
		data.depth = 1;
	}
	data.viewport = Object::cast_to<Viewport>(this);