This so-called “investigation” is a complete fraud and scandal, just like the Mueller “investigation” of the Russia hoax that this guy was involved in. The “investigator” is far more likely to face criminal charges than Dr Z is. Now that we have solid proof that the Flynn charges were knowingly false, the whole scandal is going to unravel and there will be FBI agents in prison.
The bottom line is that even if Dr Z had knowingly and deliberately claimed to be running an FDA-approved study, it would not be illegal because it was in a private communication. The government cannot regulate what individuals say to each other in a non-commercial setting. Thus there is no basis for any investigation. Now if he’d advertised such a claim, in some form that could be considered commercial speech, then there would be something to talk about, but that’s not what happened.