session_write_close has not one thing to do with request serving on the part of your server software. ALL it does is write the current session change to whatever storage method is being used to store $_SESSIONS.
Calling it will NOT have any impact on "other tasks" or release any execution back to the system. It's just a "save current $_SESSION values to file and prevent future changes to the file for this execution." -- NOTHING more, nothing less.
Has nothing to do with closing out the current "request", script execution time, or any other such nonsense.