Let's imagine Alice wants to send multiple transactions to Bob. Both Alice and Bob have open channels with Carol, not with each other:
Alice <-> Carol <-> Bob
So Alice first sends 0.01 BTC to Bob. She sets up the multi-sig tx and sends it off to Carol, which then also Bob sees. Then Alice makes another transaction to Bob of 0.02 BTC, and sets an earlier time-lock in order to invalidate the earlier 0.01 tx.
Now both Alice and Bob have seen these 2 transactions and they both go offline. They are sure that Alice has given Bob 0.02 BTC.
My question is: how do they know that Carol will commit to the blockchain (when the moment comes) the second transaction rather than the first? Couldn't Carol simply send the 0.01 tx to the blockchain rather than the 0.02 one?
I understand that Bob could come online and post it himself, and because of the timelock, he could do it before the other one, thus invalidating it.
But in the scenario where both Alice and Bob are offline do they have to trust Carol to behave honestly?