The E-infinity refinement conjecture for the DGAs $S_{2p^l-2}$

Let $p$ be the prime appearing in the definition of the DGAs $S_{2n}$, and let $l$ be a positive integer. The E-infinity refinement conjecture. For each $l>0$, the DGA $S_{2p^l-2}$ admits the structure of an $\bE_\infty$-DGA. The preceding results show that these DGAs are $\bE_\infty$ after extension of scalars to $\fp$, whereas $S_{2n}$ does not admit an $\bE_2$-DGA structure when $n\neq p^l-1$; the conjecture identifies the exceptional indices where an $\bE_\infty$ refinement may exist.