Hi,
Yes the tab work well.
the search problem I have test the website https://luxor.novanta9.net/en/ (English) and https://luxor.novanta9.net/ (italian, default language)
if I use the search field at the top, for example by searching for “dd 121”, in Italian I see correctly “Detentore Easy DD 121” among the possible results, if I do the same search from the English language I should see the product in English (DD 121) but if I am not logged in I see however the product with title in Italian. if I run the search with the admin user logged in, on the other hand, I see the product correctly in English.