Moore's conjecture on rational ellipticity and homotopy exponents

Let $X$ be a finite, simply-connected $CW$-complex. Call $X$ rationally elliptic if it has finitely many rational homotopy groups, and say that $X$ has a finite homotopy exponent at a prime $p$ if some power of $p$ annihilates the $p$-torsion in its homotopy groups. Moore's conjecture. The following are equivalent:

• $X$ is rationally elliptic;
• $X$ has a finite homotopy exponent at every prime $p$;
• $X$ has a finite homotopy exponent at some prime $p$.

Equivalently, a finite, simply-connected $CW$-complex is rationally hyperbolic if and only if it has no finite homotopy exponent at any prime. This conjecture links rational homotopy growth with torsion in homotopy groups; the paper proves it for the polyhedral products considered under additional hypotheses, while the general statement remains open.