Let $X$ be a CW-complex of finite dimension.
Suppose that $\pi_k(S^n)$ is isomorphic to $\pi_k(X)$ for all $k\geq 0$, where $n>1$.
Certainly, $\dim X\geq n$. It is easy to show that $X$ has homotopy type of $S^n$ provided $\dim X \leq n+1$.
Question: is it true for any $X$ of finite dimension? If not, is there a counter-example?