Hmm, both have benefits.

The Message Outbox could show both sides (sent and received) of the dialog of the last 24h for one recipient, like in current chat apps.

I had a demo here

The CB is actually showing sent messages for all recipients.

Since one can send messages from both boxes, it makes sense to show them in both.

But the CB is certainly more prominent, so good call to at least implement it there. 👍🏻