Hi,
I think I see the problem there. The menu is creating an exact copy of the search nodes via Javascript, and places it into the main and mobile menus. This causes a problem, as multiple search instances will exist with the same ID, which is invalid.
There is a detector script implemented into the plugin, which usually resolves this issue – however in this case it does not work correctly, as there are over 3 copies of each search instance.
Can you please add temporary FTP details as well? I would like to debug the duplicate check script.
Can you also check the back-end logins, I was not able to log-in with them.