Consumers might like to buy their metal drinking straws knowing they were made with green steel.
But the vast majority of steel is used for unseen things... Like building the piling machine that built the flood defences for your city. There is no consumer pressure to go green there. Just commercial pressure to be cheap.
Companies are investigating so they are ready incase such an incentive appears. But I see nothing on the horizon.
Note that this is pretty much out of the hands of government too - if any government mandated that all steel produced in the country was green, then all steel would just be imported and local industry would die. If the government put an import tariff on non-green steel, then makers of steel products like cars would become uncompetitive and close down. If the government gave those steel users a subsidy, it would be against WTO anti-dumping rules.
There is no good solution.