Tragic Backstory
As the title suggests, I would like to propose that Codeforces keep its extra (late) registration open throughout the contest. Unlike AtCoder, where such a system is problematic due to its penalty system, Codeforces' penalty for submissions increases every minute, and it is no different to registering 2 days before the contest, so it isn't exploitable.
I don't see any reason not to have this, unless it is too much of a technical difficulty due to room allocation?
