SHORT QUESTION: Is MAJ-3CNF a PP-complete problem under many-one reductions?
LONGER VERSION: It is well-known that MAJSAT (deciding whether the majority of assignments of propositional sentence satisfy the sentence) is PP-complete under many-one reductions and #SAT is #P-complete under parsimonious reductions. It is also apparent that #3CNF (that is, #SAT restricted to 3-CNF formulas) is #P-complete, because the Cook-Levin reduction is parsimonious and produces a 3-CNF (this reduction is actually used in Papadimitriou's book to show #P-completeness of #SAT).
It seems that a similar argument should prove that MAJ-3CNF is PP-complete under many-one reductions (MAJ-kCNF is MAJSAT restricted to kCNF formulas; that is each clause has k literals).
However, in a presentation by Bailey, Dalmau and Kolaitis, "Phase Transitions of PP-Complete Satisfiability Problems", the authors mention that "MAJ3SAT is not known to be PP-Complete" (presentation at https://users.soe.ucsc.edu/~kolaitis/talks/ppphase4.ppt). This sentence does not seem to appear in their related papers, only in their presentations.
Questions: Can the proof that #3CNF is #P-complete be indeed adapted to prove that MAJ3CNF is PP-complete? Given the statement by Bailey et al., it seems not; if the proof does not carry, then: Is there a proof that MAJ-3CNF is PP-complete? If not, is there some intuition as to the difference between PP and #P with respect to this result?