The lambda-algebra cycle conjecture for higher Lannes–Zarati stems

Let $A$ be the mod $p$ Steenrod algebra, let $M$ be an unstable $A$-module, and let $\Lambda\otimes M^\#$ be the lambda-algebra complex computing $\operatorname{Ext}_A^{s,*}(M,\mathbb F_p)$. Let $\mathscr{R}_sM$ be the $s$-th Singer construction, and consider the canonical projection

$$\Lambda\otimes M^\#\longrightarrow(\mathscr{R}_sM)^\#.$$

A positive stem means a class in $\operatorname{Ext}_A^{s,s+t}(M,\mathbb F_p)$ with $t>0$. Lambda-algebra cycle conjecture. For $s>2$, every element in a positive stem of $\operatorname{Ext}_A^{s,s+t}(M,\mathbb F_p)$ can be represented by a cycle in $\Lambda\otimes M^\#$ whose image under the canonical projection is trivial.

This conjecture implies the higher-stem triviality conjecture for the Lannes–Zarati homomorphism. The source records verification in several low-degree cases and leaves the general statement open.