Benson's strong regularity conjecture for group cohomology

Let $G$ be a finite group, let $p$ be a prime number, and let $k$ be a field of characteristic $p$. A very strongly quasi-regular system of parameters is a system of parameters of the cohomology ring $H^*(G,k)$ with the stronger quasi-regularity property described in the source.

Benson's strong regularity conjecture. The cohomology ring $H^*(G,k)$ always contains a very strongly quasi-regular system of parameters.

Benson conjectured, and Symonds proved, that $H^*(G,k)$ has Castelnuovo–Mumford regularity zero, equivalently that it contains a strongly quasi-regular system of parameters. This stronger form remains open and is relevant to the structure of group cohomology and central essential cohomology.