Huang–Theriault conjecture on mod-$p^r$ hyperbolicity

Let $X$ be a simply-connected finite $CW$-complex. For a prime $p$ and $r\geq 1$, say that $X$ is mod-$p^r$ hyperbolic if the number of $p^r$-torsion summands in the homotopy groups up to degree $m$ grows exponentially with $m$. Huang–Theriault's conjecture. If $X$ is rationally hyperbolic, then $X$ is mod-$p^r$ hyperbolic for all primes $p$ and all $r\geq 1$.

The conjecture asks whether rational hyperbolicity forces exponential torsion growth at every prime and every torsion exponent. It was proposed by Huang and Theriault and is presented here as an open conjecture; the paper establishes related results for broad classes of polyhedral products.