  We make a conjecture about the whole $E_2$ page of the May spectral sequence
in terms of generators and relations and we prove it in a subalgebra which
covers a large range of dimensions. We show that the $E_2$ page plays a
universal role in the study of Massey products in commutative DGAs. We
conjecture that the $E_2$ page is nilpotent free and also prove it in this
subalgebra. We compute all the $d_2$ differentials of the generators in the
conjecture and construct maps of spectral sequences which allow us to explore
Adams vanishing line theorem to compute differentials in the May spectral
sequence.
