Let $\Sigma^{\infty}$ denote the suspension spectrum functor from pointed topological spaces (=CGWH spaces) to orthogonal spectra. As usual, a weak equivalence of spaces is a continuous map inducing a bijection on $\pi_0$ and an iso on all $\pi_i$, $i\ge 1$, for all choices of basepoint. For orthogonal spectra, I refer to the usual notion of stable equivalence, see e.g. Mandell-May-Shipley-Schwede.
Question: If $f\colon X\longrightarrow Y$ is a weak equivalence, is $\Sigma^\infty f\colon \Sigma^\infty X\longrightarrow \Sigma^\infty Y$ a stable equivalence?
Remarks: 1. This is true if $X$ and $Y$ are well-pointed, see this MO discussion.
- In general, if $X$ and $Y$ are not well-pointed, then the reduced suspension does not preserve weak equivalences. For example, take the set $\{0\} \cup\{\frac 1n\}$ which receives a weak equivalence from a countable discrete set. The reduced suspension is a map from a wedge of spheres to the Hawaiian earrings which is not a weak equivalence.
