If the Federal Reserve really wants to fight inflation, the cost for the US economy will be very high
It is understandable that the Federal Reserve wants to fight inflation: US inflation (unlike European inflation) has mostly domestic causes: rising wages, rising corporate profit margins. But if the Federal Reserve really wants to fight inflation, it would have to raise interest rates substantially: they would have to rise more than inflation. As in the early 1980s, there would be a recession, and, given debt levels and asset prices, it would be worse than in the early 1980s. The Federal Reserve certainly does not want to do that, so it is not really going to fight inflation.