The other day, a student asked me whether, if $P \ne NP$, whether any language outside of $NP$ is known to be $NP$-hard. I wasn't sure if
- This is definitely known to be true,
- This is definitely known to be false, or
- This depends on another set of complexity assumptions that do not immediately follow from $P \ne NP$ (that is, even if we knew $P \ne NP$, this would still be an open question)
None of the texts on complexity I looked into seemed to answer this question (though it is quite possible that I simply missed it). Does anyone know which of the above three is true, or know a good reference where I could look up the answer?
(Note: This earlier question is related, but I'm considering solely questions outside of $NP$ (so the existing answer doesn't really help) and am not restricting this to just the decidable languages)