FRANKFURT, March 18 (Reuters) - The world's top economies are on track for a "soft landing" but inflation has not yet been fully defeated, the head of a central bank umbrella group warned on Monday just as policymakers debate the merits of cutting interest rates from multi-decade highs.

Global inflation has more than halved in the past year and some central banks are already unwinding ultra tight policy in the hope they can preserve growth and keep their economies going after the back-to-back shocks of the pandemic and the inflation surge.

"It seems that we are on route to a soft landing," Agustín Carstens, the General Manager of the Bank for International Settlements said in a speech in Frankfurt. "Lower inflation has come at a remarkably small cost to the real economy."

Among the world's biggest central banks, the European Central Bank could be the first to move in June, soon followed by the U.S. Federal Reserve in June or July, and the Bank of England perhaps in August, according to current market expectations.

Yet some recent indicators on underlying inflation on both sides of the Atlantic have raised fears that price growth could be more persistent than some of the more benign expectations out there suggest.

"A soft landing is not guaranteed, however. Central banks’ job is not done. While inflation is lower, it is still above central banks’ targets. And there will surely be more bumps in the road," Carstens said.

He also warned that a host of factors such as de-globalisation, economic fragmentation, adverse demographic trends and the need to fight climate change, will keep prices under pressure over the medium term, requiring central banks to follow through with their pledges to curb inflation. (Reporting by Balazs Koranyi Editing by Tomasz Janowski)