Somewhere in this forum there is an old post with Olli Savolainen, that implements Ajax-y save of work in progress for the quiz. Of course, that increases total server load, and I don't know if the suggested approach still works.
Quiz
Quizzes over time limit given 0
Müzakirə kilidlənib, beləliklə ona cavab vermək artıq mümkün deyil.