The Lock Box doesn't help.

By the time they put items in the lock box, the majority of the people in the house are already in the wrong houses so what purpose would it now serve? There is no identifier to allow them to know where they originally started.

Also, when Em is asking them what numbers they had from the original box, why would she ask Hugh? He was at a different house with a presumably different box so even if he was the original, this would serve no purpose.

Finally, when Hugh and Amir (from any universe) leave with the box, where would they leave it afterwards? They seem to be acting on the assumption that there were only two houses at that point, so assuming they left it at the house across the street, where would they go afterwards and what would be the chances they arrived at a house chronologically where alternate versions of them had just done the same.