Nothing that dramatic. Employee X, you are reminded that you are not permitted to speak as to the official government position. You may talk all you want about Bigfoot, Bigfoot, Bigfoot, but you must make clear that all opinions expressed in this interview, podcast, blog, etc., represent your personal views and do not represent the official view of the U.S. Forest Service, the U.S government, or any U.S. government agency. These are standard instructions for federal employees engaging in outside employment or volunteer work or outside speaking engagements (e.g., acting as an adjunct professor at a local college). Failure to follow those instructions may result in a wide range of disciplinary actions, ranging from an oral or written reprimand to being fired, depending on the nature of the violation. So Employee X is not fired because he says "Bigfoot is real," he's fired for ignoring the instructions or directions given him by his supervisor.
I don't know of any case directly on point, but law enforcement personnel would want to remain silent to avoid being impaired during future testimony. For example, if FBI investigator Y states on live TV, "I saw Bigfoot run across the road in front of my official vehicle," that statement will be used to impeach that investigator every time she testifies. So, at most a smart investigator would say, "Well, something ran in front of my vehicle, but I was not able to identify the animal."
Not saying that either of the above is fair, but that's what would happen.
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Norse, okay, thanks, I see what you were getting at now.