If Troy is trustworthy, he'll delete the transaction history. However, there area number of attacks possible from Troy's perspective. He can keep a transaction

If Troy is trustworthy, he'll delete the transaction history. However, there area number of attacks possible from Troy's perspective. He can keep a transaction history and use it to blackmail Alice and Bob. He can simply keep coins sent to the tumbler and send out nothing. It's very important that Troy be a trusted actor.