### 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)

switch (p_notification) {
		case NOTIFICATION_PROCESS: {
			GDVIRTUAL_CALL(_process, get_process_delta_time());
		} break;
		case NOTIFICATION_PHYSICS_PROCESS: {
			GDVIRTUAL_CALL(_physics_process, get_physics_process_delta_time());
		} break;
		case NOTIFICATION_ENTER_TREE: {
			ERR_FAIL_NULL(get_viewport());
			ERR_FAIL_NULL(get_tree());
			
			if (data.process_mode == PROCESS_MODE_INHERIT) {
				if (data.parent) {
					data.process_owner = data.parent->data.process_owner;
				} else {
					ERR_PRINT("The root node can't be set to Inherit process mode, reverting to Pausable instead.");
					data.process_mode = PROCESS_MODE_PAUSABLE;
					data.process_owner = this;
				}
			} else {
				data.process_owner = this;
			}