Is $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?
Heller gives a relativized world where $BPP = EXP^{NP}$ which sits far outside of $P^{NP}$. Showing $BPP \neq EXP^{NP}$ unconditionally would in itself be considered a major breakthrough in derandomization.