On the May Spectral Sequence at the prime 2

We make a conjecture about all the relations in the $E_2$ page of the May spectral sequence and prove it in a subalgebra which covers a large range of dimensions. We conjecture that the $E_2$ page is nilpotent free and also prove it in this subalgebra. For further computations we construct maps of spectral sequences which systematically extend one of the techniques used by May and Tangora.