I read on another thread that the offer of immunity is negotiated between the sides lawyers......and the general area of information and who is implicated spelled out.....
so if the client suddenly "can't recall" or is misleading then the immunity is withdrawn.
Sounds reasonable and proper. We all have to understand that, in Congressional investigative hearings, more goes on behind the scenes than in front of the cameras.
Most witnesses are deposed ahead of time, which allows the "prosecution" to learn what each witness will testify. Then, the sequence of appearance and the questions asked can be pre-determined in order to place the testimony into a coherent pattern that leads to a conclusion.
Indeed, many witnesses who are deposed may not be called to testify -- since they don't have anything to add.
Immunity negotiations would fall into this same background.