On the other hand, I think many companies would hesitate to create lasting records of written communications that go beyond the company’s retention policy.
While it’d be extremely useful, I think it would also increase potential liability that is currently managed by purge policies.
Granted you'd have to test it without recording that, and you'd never really know for sure if you'd fully scrubbed it, only that you hadn't found a trace of it with your subsequent queries. ...now that I say that I'm sure there's a film about roughly this, delving into an AI trying to retrieve something hidden or something? The Wargames sequel perhaps?
Your point about fine-tuning it out is interesting though. As you say though, you'd have to be really sure you'd really erased it or made it "unreachable" .
Even exfiltration of API access for a few well-prompted queries could be really bad.