All the deanonymizing work done heretofore is wrong because of multi-sig transactions
I may be mis-reading that, but it looks like that is not an offline transaction, but a transaction facilitated by a payment processor. They seem to be trying to solve the problem of quickly verifying a payment.
Offline payments should allow this:
1. Alice receives money from some online payment.
2. Alice sends money to Bob via an offline payment.
3. Bob sends money to Catherine via an offline payment.
4. Catherine spends her money online.
In that scenario, Bob should not be required to ever do anything online. In other words, it should be possible for parties to only ever deal in offline transactions, without ever going online, as it is possible for people to only ever deal in paper cash.