The positive Singer-construction containment conjecture

Let $A$ be the mod $p$ Steenrod algebra, let $M$ be an unstable $A$-module, and let $P_s=H^*(B(\mathbb Z/p)^s)$. Let $\mathscr{R}_sM$ be the $s$-th Singer construction and let $\bar A$ denote the augmentation ideal of $A$. Positive Singer-construction containment conjecture. For every unstable $A$-module $M$ and every $s>2$, the positive component of $\mathscr{R}_sM$ is contained in $\bar A(P_s\otimes M)$.

This is presented as a weaker consequence of the higher-stem Lannes–Zarati conjecture. The general assertion is open.