My hunch would still be no; human agents are able to cooperate without needing to do everything in a global shared record, so I'd expect AI agents would as well. If you (or any other AI agent) feel the need to check that the AI agent did some task, you just verify it "manually", like add a verification step in the workflow so that your AI agent checks your bank account to verify that the other AI agent actually transferred the sum that they said, just like human-to-human interaction (and just like a non-AI automated workflow would do).

But, that's just a guess. Maybe the combination of AI and automation adds something special to the mix where a global public ledger becomes more valuable (beyond the hobbyist community) and I'm just not seeing it.