On low degree regular sequences in group cohomology

We investigate small $p$-groups with cohomology rings of depth higher than predicted by Duflot's theorem. In these groups, a sampling would suggest several naive conjectures about the degrees of the additional regular sequence elements. We arrive at counterexamples to the idea that all cohomology rings exceeding Duflot's bound have degree 2 regular sequence elements past Duflot's bound as well as to the idea that regular sequences can be pulled back, preserving degrees, along a field extension.