Let $\mathbb{S}$ be the sphere spectrum. We can localize the category of based spaces, $Top_*$ at a homology theory, and hence at $\mathbb{S}$.

Equipping $Top_*$ with the Quillen model structure (weak homotopy equivalences, and Serre fibrations), and Bousfield localizing at $\mathbb{S}$ gives a model structure with the weak equivalences the isomorphisms on stable homotopy groups, i.e. $\pi_*^\mathrm{st}$-isomorphisms.

Is there a good characterization of the fibrant ($\mathbb{S}$-local) objects in this model structure?

In particular, are "nice" spaces fibrant in this model structure, e.g. nilpotent spaces, simply connected spaces, etc?