What happens if monetary policy actually has no effect on inflation?

It is possible to defend the theory that in OECD countries, monetary policy no longer has any effect on inflation, since the correlations between inflation on the one hand, and unemployment rates or output gap on the other have disappeared. If monetary policies no longer have any effect on inflation and if central banks nevertheless maintain an inflation target, the consequence will be unnecessarily extreme monetary policies: far too restrictive if inflation is higher than the inflation target, far too expansionary for a long time if inflation is lower than the inflation target. To get out of this trap, and as long as inflation no longer reacts to the economic cycle, the only solution is for central banks to move to another objective, which may be: Control of long-term interest rates, as in Japan, which has been possible since central banks began using quantitative easing; Liquidity control to prevent financial instability, which is probably not their intention at present; Overheating of the real economy (as from 2016 to 2019 in the United States) to force companies to increase productivity and employment among the low-skilled. Even when the fiction of the inflation target is maintained, central banks in practice now combine policies of keeping long-term interest rates low and boosting the employment rate.