The degree-two superduflot regular sequence conjecture

Let $G$ be a finite $p$-group. Its cohomology ring $H^*(G,\mathbb F_p)$ is a graded-commutative, noetherian, local $\mathbb F_p$-algebra. Write $\operatorname{rank}_p Z(G)$ for the rank of the $p$-subgroup of the center, and call $G$ superduflot if

$$\operatorname{depth} H^*(G,\mathbb F_p)>\operatorname{rank}_p Z(G).$$

A superduflot regular sequence is the part extending a Duflot regular sequence to a maximal regular sequence; its elements beyond the Duflot bound are superduflot elements. A regular sequence has degree sequence $a_1,\dots,a_d$ when its elements have respective degrees $a_1,\dots,a_d$. The degree-two superduflot regular sequence conjecture. Every superduflot finite group has a superduflot regular sequence whose superduflot elements all have degree at most $2$. The conjecture was disproved by examples of superduflot groups whose low-degree cohomology classes have a common annihilator, so that the required degree-two elements cannot exist.