Selecting the Correct Oracle

After my post last week on the Complexity accepts, a friend of Shuichi Hirahara send Shuichi an email saying that I was interested in his paper. Shuichi contacted me, sent me his paper and we had a few good emails back and forth. He posted his paper Identifying an Honest EXP^NP Oracle Among Many on the arXiv yesterday.

Shuichi asks the following question: Given two oracles both claiming to compute a language L, figure out which oracle is correct. For which languages does there exist such a selector?

For deterministic polynomial-time selectors, every such L must sit in PSPACE and all PSPACE-complete languages have selectors. The question gets much more interesting if you allow probabilistic computation.

Shuichi shows that every language that has a probabilistically poly-time selector sits in S₂^EXP, the exponential analogue of S₂^P. His main theorem shows that EXP^NP-complete sets have this property. His proof is quite clever, using the EXP^NP-complete problem of finding the lexicographically least witness of a succinctly-described exponential-size 3-SAT question. He uses PCP techniques to have each oracle produce a witness and then he has a clever way to doing binary search to find the least bit where these witnesses differ. I haven't checked all the details carefully but the proof ideas look good.

Still leaves an interesting gap between EXP^NP and S₂^EXP. Is there a selector for Promise-S₂^EXP-complete languages?