This question is closely related to my previous question about modules over truncated sphere spectra, in particular, it has the same motivation.

Recall that every space (or ∞-groupoid) can be represented as the homotopy colimit of some simplicial diagram of 0-truncated spaces, i.e., sets considered as homotopy 0-types.
For example, if X is a Kan simplicial set, then the homotopy colimit of X considered as a functor X: Δ^{op} → Set → Space is equivalent to X itself.

For connective spectra (i.e., group-like E_{∞}-spaces) a more complicated picture emerges:
homotopy colimits of simplicial diagrams of 0-truncated connective spectra (i.e., abelian groups) are precisely Eilenberg-MacLane spectra (of connective chain complexes of abelian groups).
More generally, homotopy colimits of simplicial diagrams of k-truncated connective spectra for some k>0 are modules over
the k-truncation π_{≤k}S of the sphere spectrum S.
In particular, not every connective spectrum can be represented in this way.
(Presumably, being a module over π_{≤k}S is also a sufficient condition, but I haven't checked any details to claim this.)

I wonder what happens in the intermediate case of (not necessarily group-like) E_{∞}-spaces.

**Which E _{∞}-spaces can be represented as homotopy colimits of k-truncated E_{∞}-spaces for some k>0? What if we require the diagrams to be simplicial, i.e., indexed by Δ^{op}?**

In particular, if some E_{∞}-spaces cannot be represented in this way, what tools do we have to detect this?

Specifically, I'm interested in the answer for the case of E_{∞}-spaces coming from connective spectra like MU, MSO, KU, or KO.