### 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)

ERR_FAIL_COND_MSG(data.inside_tree && !Thread::is_main_thread(), "Removing children from a node inside the SceneTree is only allowed from the main thread. Use call_deferred(\"remove_child\",node).");
	ERR_FAIL_NULL(p_child);
	ERR_FAIL_COND_MSG(data.blocked > 0, "Parent node is busy adding/removing children, `remove_child()` can't be called at this time. Consider using `remove_child.call_deferred(child)` instead.");
	ERR_FAIL_COND(p_child->data.parent != this);
	/**
	 *  Do not change the data.internal_children*cache counters here.
	 *  Because if nodes are re-added, the indices can remain
	 *  greater-than-everything indices and children added remain
	 *  properly ordered.
	 *
	 *  All children indices and counters will be updated next time the
	 *  cache is re-generated.
	 */
	data.blocked++;
	p_child->_set_tree(nullptr);
	
	remove_child_notify(p_child);
	p_child->notification(NOTIFICATION_UNPARENTED);
	data.blocked--;
	data.children_cache_dirty = true;
	bool success = data.children.erase(p_child->data.name);
	ERR_FAIL_COND_MSG(!success, "Children name does not match parent name in hashtable, this is a bug.");
	p_child->data.parent = nullptr;
	p_child->data.index = -1;
	notification(NOTIFICATION_CHILD_ORDER_CHANGED);
	emit_signal(SNAME("child_order_changed"));
	if (data.inside_tree) {
		p_child->_propagate_after_exit_tree();
	}
}