Oooh! Ooooh! Idea!
The US has a double jeopardy technicality that allows for a charge already ruled in a criminal trial to be raised again in a civil trial. For example in the OJ Simpson trial he was found not guilty in a criminal trial but the plaintiffs then raised a civil trial in which he was forced to pay damages. Where this could come into play in the LCS is when the justice system becomes liberal enough, people start getting forced into rehabilitation rather then imprisonment. I don't think it's beyond plausible to imagine that this could be done through a civil court, so long as it's forced rehabilitation rather then prosecution.
The way this plays out is that when a crime is added to your rap sheet it is also added to a second, hidden rap sheet. Then when society becomes liberal enough, you get summoned to court to face psychiatric evaluation to find out if you are a threat to yourself and others. This court would probably have some rules a bit different from the other court to gear it towards end game challenge. But most importantly, if you beat this court, it clears your name but not your secondary rap sheet. There wont be a warrant for your arrest until you commit new crimes, but every time you are arrested for new crimes, your entire criminal history will weigh against you in the psychiatric hearing.
Furthermore, by entire criminal history, I mean entire criminal history, including the jaywalking and speeding tickets you racked up before the LCS. Your permanent record from highschool? It's there. Even if he keeps his nose completely clean in the LCS, you might get your founder sent to rehabilitation for setting off the fire alarms to skip a algebra test...