The Z_2 periodicity conjecture

Let $X$ be a connected topological space. A nonzero element $x\in H^k(X;\mathbb{Z}_2)$ induces periodicity if multiplication by $x$ gives isomorphisms

$$H^i(X;\mathbb{Z}_2)\longrightarrow H^{i+k}(X;\mathbb{Z}_2)$$

for all $i\geq 0$; in this case $H^*(X;\mathbb{Z}_2)$ is $k$-periodic. The minimum period is the least such $k\geq 1$. The $\mathbb{Z}_2$ periodicity conjecture. If $H^*(X;\mathbb{Z}_2)$ is $k$-periodic and $k$ is the minimum period, then $k\in\{1,2,4\}$. This conjecture extends Adams' theorem from singly generated cohomology to periodic cohomology. It is known that the minimum period is a power of two, but the restriction to $1$, $2$, or $4$ remains open.