The mod-p^r hyperbolicity amplification of Moore's conjecture

Let $p$ be a prime and $r$ a positive integer. A $p$-local space $X$ is mod-$p^r$ hyperbolic when the number $t_n$ of $\mathbb{Z}/p^r$-summands in $\bigoplus_{m\leq n}\pi_m(X)$ grows exponentially, namely

$$\liminf_n\frac{\ln t_n}{n}>0.$$

The mod-$p^r$ hyperbolicity conjecture. Let $X$ be a finite simply-connected $CW$-complex. If $X$ is rationally hyperbolic, then it is mod-$p^r$ hyperbolic for all primes $p$ and positive integers $r$. The conjecture strengthens Moore's conjecture by requiring exponential growth of every prescribed torsion order, rather than merely torsion of arbitrarily high order. The paper's preceding results establish this behavior for certain spaces and, after excluding finitely many primes, in a broad family, but the stated universal claim remains open.