I logged in this morning, and I noticed that my "Search with..." list had 2 entries of each site, and I manually fixed this by deleting each of the duplicate entries.
However, I carried out additional searches (changing the region occasionally) and now my "Search with..." dropdown appears as the screenshot below.

I'm not sure if it was related, but I had triggered the bug https://kagifeedback.org/d/3387-no-search-results-found in my searches.
The list of "Search with..." entries should not grow unless I add additional elements to it.