Formality of the GL-operad for supersolvable arrangements via MD and OS

Establish the formality of the GL-operad of Fulton–MacPherson compactifications restricted to supersolvable complex hyperplane arrangements by developing a method that leverages the GL-cooperad structure on MD and the quasi-isomorphism MD(L) \xrightarrow{\sim} OS(L) for supersolvable geometric lattices L.

Background

Proposition \ref{propcoop} constructs a GL-cooperadic structure on MD and shows compatibility with OS, while Theorem \ref{theoqiso} proves MD(L) ≃ OS(L) for supersolvable L. In the braid case, analogous structures underlie formality results.

The authors suggest these ingredients should imply formality for the GL-operad in the supersolvable case but explicitly note that they do not see how to carry out the proof with the current tools.

References

In principle Proposition \ref{propcoop} together with Theorem \ref{theoqiso} should lead to a proof of the formality of the $GL$-operad of Fulton--Macpherson compactifications restricted to supersolvable complex hyperplane arrangements, in analogy with the braid case, but it is not clear to the author how one would use this result.

— Matroid complexes and Orlik-Solomon algebras  (2506.15048 - Coron, 18 Jun 2025) in Section 5 (Additional structures on MD(L)), closing paragraph after Proposition \ref{propcoop}