### 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)

for (HashMap<StringName, Node *>::Iterator I = data.children.last(); I; --I) {
		I->value->_propagate_exit_tree();
	}
	data.blocked--;
	GDVIRTUAL_CALL(_exit_tree);
	emit_signal(SceneStringNames::get_singleton()->tree_exiting);
	notification(NOTIFICATION_EXIT_TREE, true);
	if (data.tree) {
		data.tree->node_removed(this);
	}
	if (data.parent) {
		Variant c = this;
		const Variant *cptr = &c;
		data.parent->emit_signalp(SNAME("child_exiting_tree"), &cptr, 1);
	}
	
	for (KeyValue<StringName, GroupData> &E : data.grouped) {
		data.tree->remove_from_group(E.key, this);
		E.value.group = nullptr;
	}
	data.viewport = nullptr;
	if (data.tree) {
		data.tree->tree_changed();
	}
	data.inside_tree = false;
	data.ready_notified = false;
	data.tree = nullptr;
	data.depth = -1;
}
void Node::move_child(Node *p_child, int p_index) {