The title pretty much says it all, but to explain how I got there:
I think, that one of the reasons we are unable to prove or disprove , but mainly disprove $P=NP$ (and yes, I was provoked by the discussed recent proof about $P$ vs $NP$) is because we can't take some superpolynomial subexponential problem and reduce it to some $NP$-complete, because in this "grey" area above $P$ are two types of problems - ones in $NP$ and ones not in $NP$. But I know little to nothing about these problems, nor what is the state-of-the-art of current scientific community knowledge, but I just wonder whether there is a problem known to have this property.
I\ve been going through similar question to this one, but I could not deduce from posts there answer to this question.
EDIT: It is irrelevant whether the language is mathematically artificial or describes something "natural", as long as one can prove it is in $QP$ and not in $NP$, or at least give a very convincing argument for it.