I think what happens is you are presented with two tournament games for each opponent (1W-1B). The timeout clock starts running then for BOTH games (3 days, etc.). If your opponent has not made his white move in the timeout period, then his timebank amount, if any, starts being used. When that has been depleted, the skull shows up and you may claim the win. If you do not claim the win after 2(?) days, it is done automaticly to help speed up the tourney.
I think when your opponent has yet to make the first move the clock is running, but you can't see it for his game as white.
In any case don't offer the draw.