Mahowald's conjecture on periodic classes in the cohomology of finite sub-Hopf algebras

Let $A(j)$ be the finite sub-Hopf algebra of the mod-$2$ Steenrod algebra indexed by $j$, and let $v_n$ denote the corresponding periodic cohomology class. For natural numbers $n$ and $k$, consider the class $v_n^{2^{n+1+k}}$ in the cohomology of $A(j)$.

Mahowald's conjecture. For any natural numbers $n$ and $k$, the class $v_n^{2^{n+1+k}}$ is defined and nonzero in

$$\operatorname{Ext}_{A(j)}(\mathbb F_2,\mathbb F_2)$$

for all $j=n,n+1,\dots,2n+k$.

This conjecture concerns the persistence of periodic classes through finite sub-Hopf algebras of the Steenrod algebra. It was originally due to Mahowald around 1980 and first appeared in print in work cited by the source; the paper states that proving it is the goal of the note. The supplied text gives no resolution status.