Hu'ng–Peterson algebraic conjecture on spherical classes

Let $A$ be the mod-$2$ Steenrod algebra and let $D_k$ be the Dickson algebra. The Lannes–Zarati homomorphism is

$$\varphi_k:\operatorname{Ext}_A^{k,k+i}(\mathbb Z/2,\mathbb Z/2)\longrightarrow (\mathbb Z/2\otimes_A D_k)^*_i.$$

Algebraic conjecture on spherical classes. The homomorphism $\varphi_k$ is zero for all $k>2$.

The conjecture is motivated by the fact that Hopf invariant one and Kervaire invariant one elements are detected on the first and second lines of the Adams spectral sequence. The paper explicitly says that this conjecture remains open, although vanishing on a permanent cycle need not imply vanishing of the Hurewicz image.