The Z_p periodicity conjecture

Let $X$ be a topological space and let $p$ be an odd prime. A nonzero element $x\in H^k(X;\mathbb{Z}_p)$ induces periodicity if multiplication by $x$ gives isomorphisms

$$H^i(X;\mathbb{Z}_p)\longrightarrow H^{i+k}(X;\mathbb{Z}_p)$$

for all $i\geq 0$. Assume that $x$ has minimal degree among all such elements. The $\mathbb{Z}_p$ periodicity conjecture. Then $k=2\lambda$ for some divisor $\lambda$ of $p-1$. The cited periodicity theorem establishes only that $k=2\lambda p^a$ for such a divisor $\lambda$ and some $a\geq 0$; the conjecture is that $a=0$.