Hi,
Thank you.
The same navigation is present (your screenshot), on both the staging and the live – on the staging there are more items to display, so it is displayed. It can be turned off here: https://i.imgur.com/uJn102z.png
I also found the issue, it was due to this option was disabled: https://i.imgur.com/PCRMe3U.png
The live server has a much older version of the search, where this feature worked a bit differently as the current one. I turned that option on on the staging site, as it makes no difference in layout in your case.
It should be all right now on the staging as well.