Moore's uniform torsion conjecture for finite simply-connected spaces

Let $X$ be a simply-connected finite space. Say that $X$ has a uniform $p$-torsion bound if there is an integer $N$ such that the $p$-primary torsion in every homotopy group of $X$ is annihilated by $p^N$. Moore's conjecture. The space $X$ has a uniform $p$-torsion bound on its homotopy groups if and only if

$$\pi_*(X)\otimes\mathbb{Q}$$

is finite dimensional. The conjecture is raised as a question about the optimality of the torsion bounds discussed in the source; no resolution is supplied.