I think you're probably chewing on this because it feels similar to a ZKP setup. But ZKPs don't really work in meatspace, as meatspace lying is not perfect. Thus, your belief in the validity of the statement will leak, and so telling you can never actually be zero-knowledge.

A proper ZK analog would be the company being prohibited from giving you a signed statement saying they had said coverage, as if you were kidnapped the kidnappers would force you to turn over this signed statement to know you represent a nice payday. So they type out a statement, then sign it with an X in front of you. If the kidnappers see this, they cannot be sure you didn't just type up the statement and sign it with your own X - only you know it was signed by the company.

But back in the physical world your kidnappers will just "prove" the fact to themselves by beating it out of you. Which is why the K&R policy prohibits any sort of telling you, not just written - your knowing is the liability.