Is the following claim true in general?
Claim: A map $f:(X^r)\rightarrow (Y^r)$ of $n$-connective spectra (of simplicial sets) is a stable equivalence if and only if the corresponding map $f:X^n\rightarrow Y^n$ is a weak equivalence.
The original claim is for $n=1$, although I don't know why. Any explanation will be appreciated.