Connecting Course Exit to Browser Closing - or Establishing Exit Time

I have a test locked down (both the slides and the player).  I have a timer restricting the amount of time allowed in the test.  The questions are attempted only once, no access to previous button, no retry.  I would like to connect the closing of the browser to an end of attempt with the test.  Is this possible?  The goal would be that if a user closes the browser window, the test is marked as submitted results/completed, and they cannot go back into the test and resume where they were.  Right now, they can close the browser window, then open the module which returns them back to the test question they were in at the time they closed the browser and with the timer set back to the time they first launched that test question.  (I don't want them to gain that time back.)  If I could keep them from regaining that time, I wouldn't have to worry about the browser being closed.