I was not able to find any particular tradition from authentic sources.
12th Muharram they arrived at Kufa. It took them two days.
Kufa then is Najaf right now. A book called Sa’aadatud daarayn fee maqtalil Husayn by Ayatull….Najafi contains all famous/reliable/authentic references and such an incident is not cited there.
Saqalain Abbas Alawi