Characterization of Pierce and pure ideals of Green functors

Determine an explicit characterization of the Pierce ideals and pure ideals of an arbitrary Green functor \(\underline{R}\in CAlg(\mathrm{Sp}_G^\heartsuit)\), and determine whether analogous descriptions exist for the other projectively rigid examples listed in the paper.

Background

For connective E∞\mathbb{E}_\infty-rings, the paper identifies atomic, Serre, and general smashing frames with Pierce, pure, and idempotent ideals of π0R\pi_0R. It asks for a comparable algebraic description in the equivariant Green-functor setting and in the other examples of projectively rigid algebras.

References

Let $G$ be a finite group. Is there an explicit characterization of the Pierce ideals or pure ideals of an arbitrary Green functor $\underline{R}\inCAlg(_G\heartsuit)$? Do similar descriptions exist for the other examples in \cref{exalgttt}?

— Generalized Telescope Conjecture  (2609.03375 - Liang, 3 Sep 2026) in Section 7, subsection “Over a connective $\mathbb{E}_\infty$-ring,” final question