Adem's persistence conjecture for torsion in finite-group cohomology

Throughout, let $H^i(S)=H^i(S;\mathbb{Z})$ denote the integral cohomology of a finite group $S$. For a finite $p$-group $P$, let $e(P)$ be the exponent of $\bar{H}(P)=\bigoplus_{i=1}^{\infty}H^i(P)$, and let $e_{\infty}(P)$ be the least positive integer $n$ such that $n\bar{H}(P)$ is finite. Adem's conjecture. If $H^i(S)$ contains an element of order $p^n$ for some $i$, then it contains an element of order $p^n$ for infinitely many $i$. In particular, for a finite $p$-group $P$,

$$e_{\infty}(P)=e(P).$$

The conjecture was stated as part of a question about whether the eventual exponent of the cohomology agrees with its full exponent. The paper gives a counterexample: a finite $p$-group has elements of order $p^3$ in some integral cohomology groups, while $p^2$ annihilates all sufficiently high cohomology groups, so the conjecture is false in general.