Determine how widely Morava K-theories admit E∞-algebra structures over other E∞-rings

Determine for which E∞-rings E the Morava K-theory spectra K(n) (and, more generally, similar E∞-rings) admit the structure of E-algebras; equivalently, characterize the class of E∞-rings E for which there exists a unital E∞-algebra map E → K(n).

Background

The authors’ redshift results for Lubin–Tate theories E(k,G) and related spectra raise structural questions about how Morava K-theories interact with other E∞-rings via algebra structures.

Understanding for which ambient E∞-rings the spectra K(n) can be realized as E-algebras would clarify how broadly these spectra arise within the landscape of highly structured ring spectra and could impact strategies for proving redshift and related properties in greater generality.