Moore's conjecture on rational hyperbolicity and homotopy exponents

Let $X$ be a simply-connected finite $CW$-complex. A space is rationally hyperbolic when its rational homotopy groups grow exponentially, and $cexp_p(X)$ denotes its homotopy exponent at the prime $p$, with $cexp_p(X)=cinfty$ when the $p$-torsion in $cpi_*(X)$ has unbounded order.

Moore's conjecture. The following are equivalent:

$$X\text{ is rationally hyperbolic};$$ $$\exp_p(X)=\infty\text{ for some prime }p;$$ $$\exp_p(X)=\infty\text{ for all primes }p.$$

The conjecture connects the rational elliptic–hyperbolic dichotomy with the existence of torsion homotopy groups of arbitrarily high order. It is known for spheres and for mod-$p^r$ Moore spaces when $p>2$ or $r>1$, but remains open for mod-$2$ Moore spaces and is otherwise unresolved in general.