The strict inequality conjecture for primitive detection degrees

Let $G$ be a finite group. Let $e(G)$ be the degree associated with the top primitives, and let $e^{\prime}(G)$ be the largest degree $d$ for which $Q_A Cess^d(G)$ is nonzero when $Cess^*(G)$ is nonzero; if $Cess^*(G)=\mathbf 0$, set $e^{\prime}(G)=-1$.

Strict inequality conjecture. If $G$ is not $p$-central, then

$$e^{\prime}(G)<e(G).$$

This conjecture compares the primitive detection degree with the top primitive degree. The source proves it when $r-c=1$ and explains that the Strong Regularity Conjecture implies it for a fixed finite group; Benson's conjecture is known when $r-c\leq 2$, but the general status remains open.