Adams-type periodicity conjecture for mod-$p$ cohomology

Let $p$ be a prime, let $M$ be a topological space, and let $x\in H^k(M;\mathbb{Z}_p)$ be nonzero. Say that $x$ induces periodicity up to degree $r$ when multiplication by $x$ gives the periodicity maps used in the paper through degree $r$. Assume that $x$ induces periodicity up to degree $pk$ and has minimal degree among all such elements. Adams-type periodicity conjecture.

$$\begin{cases} p=2:\quad k\in\{1,2,4,8\},\quad\text{and if $x$ induces periodicity up to degree $3k$, then } k\neq 8;\\ p>2:\quad k=2\lambda\text{ for some }\lambda\mid p-1. \end{cases}$$

This would strengthen the periodicity theorem by imposing the restrictions obtained by Adams for singly generated cohomology rings. The source does not establish the conjecture, so its general validity remains open.