Fix a prime $p$. I have a sketch of a proof that if $X$ is a finite simply-connected CW complex with $\mathrm{dim}(X) < p$ then for some $t\in \mathbb{N}$, the $p$-localization $\Sigma^t X_{(p)}$ is a wedge of Moore spaces.

(Basically, the idea is that all the interesting attaching maps are Whitehead products, hence stably trivial.)

Questions:

- Does anyone know a reference for this?
- If it's not true, I'd love to know that too!

EDIT: Perhaps this is just a theorem of tame homotopy theory?