You're talking almost 60 years ago!!I think you misunderstand what "conclusive" means. Something can be "conclusive" and still *wrong*.
The courts believed they were being presented with "conclusive evidence" that the Gilford 4 and Birmingham 6 were unequivocally guilty.
As such they carried out sentencing, which if capital punishment had been a thing would have undoubtedly resulted in the death sentence for most, if not all of them.
That "conclusive" evidence turned out to be false / fraudulent / incorrect.