The Singer-transfer Lannes–Zarati conjecture

Let $A$ be the mod $p$ Steenrod algebra, let $E_s$ be the elementary abelian group of rank $s$, let $\operatorname{Ann}(H_tBE_s)$ denote the relevant annihilator subspace, and let $[\,\cdot\,]_{GL_s}$ denote coinvariants under $GL_s$. Let $\varphi_s$ be the Lannes–Zarati homomorphism and $Tr_s$ the Singer transfer, and define

$$j_s:=\varphi_s\circ Tr_s:[\operatorname{Ann}(H_tBE_s)]_{GL_s}\to \operatorname{Ann}(\mathscr{B}[s]^\#)_t.$$

The Singer-transfer Lannes–Zarati conjecture. The composition $j_s$ vanishes in every positive degree $t$ for $s\geq 3$. This is presented as a weak version of the graded-associated Lannes–Zarati conjecture, restricting the map to the image of the Singer transfer; the source gives no resolution status.