The augmentation-ideal conjecture for the Singer-transfer complex

Let $A$ be the mod $p$ Steenrod algebra, let $E_s$ be the elementary abelian group of rank $s$, and let $\mathscr{B}[s]$ be the subobject defined in the paper inside $H^*BE_s$. Write $\bar A$ for the augmentation ideal of $A$. The augmentation-ideal conjecture. For every $s\geq 3$,

$$\mathscr{B}[s]\subset \bar A H^*BE_s.$$

The paper states this as an equivalent formulation of the weak Singer-transfer conjecture, via the chain-level description of the dual map; the source gives no resolution status.