Cash Shuffle is a powerful tool for cycling a coin through many joined transactions. However, after shuffling a wallet, a user will inevitably wish to consolidate coins, and for this another tool is needed.
We need a method to coordinate coinjoin transactions with multiple inputs per user. This is inherently challenging because we want to hide input linkages while simultaneously attempting to blame/ban users who don't sign all their inputs.
This scheme takes a "sharding" approach whereby each player gives each other player only 1 input to verify.
(Assume we have 10 players using 9 outputs each). If it can be successfully implemented, this would be an improvement over schemes that trust the servers with information about linkages between inputs. In the long run, that trust could create pressure and incentives for bad actors to compromise servers.