I wonder1 whether there is a known relativization barrier against proving $L\neq NP$. Hence I'm looking for a language $A$ for which $L^A=NP^A$.
My first idea was to try $A:=SAT$, but then I thought that $L^A\subset P^A = \Delta_2^P$ and $NP^A=\Sigma_2^P$. This seems to disqualify not only $A:=SAT$, but any complete problem $A$ from the polynomial hierarchy for my purpose.
My next idea was to try $A:=TQBF$, where $TQBF$ is the $PSPACE$-complete problem to decide true quantified Boolean formulas. But $P^A=NP^A$ is well known, and $L^A=NP^A$ is a stronger statement, so $L^A=NP^A$ would be well known, if anybody had proved it.
My question is just whether there is a known relativization barrier. We certainly can't prove that such a language $A$ can't exist, because otherwise we would get $L\neq NP$ as corollary.
1. I know that logarithmically space bounded TMs with stack recognize exactly the languages from $P$, independent of whether the TM is deterministic or not. I don't know whether this result relativizes, but I guess it does. So I wonder whether there can be a language $A$ with $NL^A=P^A$. But asking for $L^A=NP^A$ instead seems easier, because of the connection to the polynomial hierarchy.