"Largest" class properly contained in PSPACE for a random oracle Green [1] showed that $PP^{PH}$ is properly contained in $PSPACE$ relative to some oracle. Around the same time, in the famous "voting polynomials" paper [2], it was shown that $PP$ is properly contained in $PSPACE$ for a random oracle.

Question


  Is it also true that $PP^{PH}$ is properly contained in $PSPACE$ relative to a random oracle?


Observations


From what I understand, this can be proved by showing that "small" $AC^{0}$ circuits with a MAJORITY gate at the root fail to compute the PARITY function even on a (1/2+small)-fraction of inputs. I found such claim in Theorem 6 of [3]. Does this reasoning make sense? Am I interpreting Klivans' theorem in the right way?
I wrote "largest" class in the title, because I am also interested in classes that have this same property and that are uncomparable with $PP^{PH}$.


Motivations


Nostalgia for 90s research :)
I am looking at similar questions in quantum complexity theory.


References


Frederic Green, An oracle separating $\oplus P$ from $PP^{PH}$, Information Processing Letters, '91
James Aspnes, Richard Beigel, Merrick Furst, and Steven Rudich, The expressive power of voting polynomials, STOC '91
Adam Klivans, On the Derandomization of Constant Depth Circuits, RANDOM '01
Yes, because $PP^{PH}=PP$ relative to a random oracle. Follows from Toda-Ogihara