User blog comment:Primussupremus/Extended busy beaver or super sigma./@comment-29915175-20170406195949/@comment-1605058-20170407151419

I doubt any such kind of relation will be proven any time soon, since (at least naively) this requires us to know that if some machine works for a given number of steps, then there is a machine with the same number of states which writes a not much smaller number of ones. This means that somehow no machine writes ones "terribly inefficiently", so that it writes few ones in a large number of steps, but no machine can do better. I feel like it can't be excluded without knowing the behavior of all these machines.