Completely false. Corinth was a Roman colony in Greece. The Church there had been established by Paul who spent his last years communicating from Rome. Clement was a close companion of Paul's and was well known to the Corinthian Church. They were not in communication with the Apostle John as he was over in Ephesus or perhaps by then was exiled to Patmos. The circumstances in no way suggest they considered the Bishop of Rome to be above the Apostle John.
This is the primary flaw in all Catholic apologetics, that of conflating St Peter with the papacy. If St John Chrysostom says something special about St Peter then they assume he means the same for the bishop of Rome. However when St John Chrysostom says something special about one of the other apostles, it is of no importance.
For example, St John Chrysostom states that the Apostle John is the holder of the keys. Catholic apologists ignore that.
The very fact that Chrysostom received his ordination from bishops who were not in communion with Rome, and he wrote nothing at the time which suggested he felt communion with Rome was necessary, demonstrates that whatever he wrote about St Peter, referred to St Peter alone. His writings nowhere show he believe that those things somehow transferred and became true of the Bishop of Rome