How far can Spectral Algebraic Geometry be developed over $\mathbb{E}_2$-rings (instead of $\mathbb{E}_\infty$-rings)?

Jacob Lurie has extensively developed derived algebraic geometry in the setting of $ \mathbb{E}_\infty$ -ring spectra [SAG]. The resulting theory of Spectral Algebraic Geometry (SAG) gives (in particular) a way to view some of the spectra topologists care about algebro-geometrically, as spectral schemes. In particular, examples of such spectra would be

  1. The complex cobordism spectrum $ \mathrm{MU}$ ;

  2. Complex $ K$ -theory $ \mathrm{KU}$ ;

  3. The spectrum $ \mathrm{TMF}$ of topological modular forms;

  4. The sphere spectrum $ \mathbb{S}$ .

On the other hand, there are many important spectra which don’t admit $ \mathbb{E}_\infty$ -structures (and hence don’t fit in Lurie’s SAG) such as:

  1. The $ p$ -local Brown-Peterson spectrum $ \mathrm{BP}$ and its connective covers $ \rm{BP}\langle n\rangle$ ;

  2. The Morava $ K$ -theories $ K(n)$ ;

  3. The Ravenel spectra $ X(n)$ .

There has been work on SAG over $ \mathbb{E}_n$ -rings, in particular by John Francis, in his thesis. Focusing on $ n=2$ , what results of SAG are expected to be troublesome to extend to the setting of $ \mathbb{E}_2$ -ring spectra?

I’m also tempted to ask here Sanath’s question on this topic:

[…] what are some results in either the purely algebro-geometric or purely chromatic aspects of spectral algebraic geometry which rely upon using the entire $ \mathbb{E}_\infty$ -ring structure?

to be found in his A love letter to E_2-rings.