The graded associated Lannes–Zarati conjecture

Let $p$ be an odd prime, let $A$ be the mod $p$ Steenrod algebra, and let $\varphi_s$ denote the mod $p$ Lannes–Zarati homomorphism in Adams filtration $s$, with positive stem $t$. The graded associated Lannes–Zarati conjecture. The homomorphism $\varphi_s$ vanishes in every positive stem $t$ for $s\geq 3$. The paper presents this as a graded-associated analogue of the classical conjecture on spherical classes, motivated by its results for $s\leq 3$; it is not claimed there to imply the classical conjecture, and the source gives no resolution status.