Speedup for Natural Problems and NP=?coNP

Informally, a language L has speedup if, for any Turing machine (TM) for L, there exists one that is better. Blum [2] showed that there are computable languages that have almost-everywhere speedup. These languages were unnatural in that they were constructed for the sole purpose of having such speedup. We identify a condition apparently only slightly stronger than P 6= NP which implies that accepting any coNP -complete language has an infinitely-often (i.o.) superpolynomial speedup and NP 6= coNP . We also exhibit a natural problem which unconditionally has a weaker type of i.o. speedup based upon whether the full input is read. Neither speedup pertains to the worst case.