Generation conjecture for genuine equivariant stable cohomology operations

Let $\Gmr$ be a finite group and let $p$ be a prime. For each $i$, let $\Ecal^{(p^i)}_{\Gmr,p}$ be the collection of integral $\uprho_{\Gmr}$-stable $\Hmr\underline{\FF}_p$-Eulerian sequences of weight $p^i$, and write

$${\bf S}_{\Gmr,p}:=\{\Sfrak^{\upchi}:\upchi\in\bigsqcup_i\Ecal^{(p^i)}_{\Gmr,p}\}.$$

Let $\Acal_{\Gmr,p}^{\star}$ denote the algebra of genuine stable $\uprho_{\Gmr}$-stable cohomology operations. Generation conjecture. The collection

$${\bf S}_{\Gmr,p}$$

generates $\Acal_{\Gmr,p}^{\star}$ for every finite group $\Gmr$ and every prime $p$. This predicts that all genuine stable cohomology operations in the stated equivariant setting are generated by operations associated to Eulerian sequences of $p$-power weights; the source gives no resolution of the conjecture.