I don't think so, bitcoin price will not affect the fiat, even the price of bitcoin reach $100.000, fiat still worth and many people still using fiat. Bitcoin will not affect fiat.
yes it is not such an easy job to make fiat worthless, actually the number of people using bitcoin is very low therefore most of the people are still using and depending of fiat, therefore fiat cannot be made worthless in recently.
The fiat will always be supported by the government. That is a good way to exploit the people with inflation.