I recently do often search queries like title:problem [java] closed:0 on Stack Overflow.
These give quite a lot of results, and are in no particular order ("Relevance", whatever this means). Thus I select the "active" tab to see recently active questions with these search terms (to avoid spamming the first page when editing too old questions). I click some questions (and edit the titles).
Then I think "Are there also any new questions if I change problem to problems"? So, I add an s to the edit box over the list, press Enter, and ...
I'm again on the "relevance" tab (with the new search term). So I have to click the "active" tab again.
I think that I did this at least a dozen times today (with slightly differing search terms: problem, question, help, problems, questions, and in the tags java, bash, git, javac, javadoc, encryption, cryptography, ..., with and without closed:0).
Could the search, at least if invoked from this search box over the result list, please remember on which tab I was, and show the result in the same tab (e.g. ordered the same way)?