Obviously, you already knew the answer before yo posed the question. Even a broader, "how many have coaches have failed to improve a program by their 5th season and then achieved a high level of success?" query and we all know that such results are extreme exceptions.
To be fair, a main reason it is extraordinarily unusual for it to happen is largely because it is extraordinarily unusual for a coach to be given the opportunity.
To get any meaningful historical information, we need to have a sample of a significant number of coaches who were allowed to continue coaching at the same school for well beyond 5 years despite the lack of improvement for the first 5, and then look at how many made a later improvement and how many didn't.
That would be a lot of work to try to gather such information when it seems highly probable that all it would lead to is the conclusion that the sample size is way too small to mean anything.
Of course, it also means that continuously throwing Beamer's name in to defend Holgorsen is equally worthless. One example from a quarter century ago is insignificant.