Invariant cell numbers for algebras over infinity-operads

Determine for what class of infinity-operads \(\mathcal O\) under \(\mathbb E_1\) the \(\mathcal O\)-algebras in spectra automatically satisfy the invariant cell number property.

Background

The paper proves that every non-zero $\mathbb E_4$-ring has the invariant cell number property by exploiting the stably $\mathbb E_3$-monoidal structure on its module category and constructing Schur-functor operations. The authors explain that their techniques do not extend directly to $\mathbb E_2$- or $\mathbb E_3$-rings: the relevant power operations in the $\mathbb E_2$-monoidal setting are governed by representations of braid groups, whose representation theory is described as much less well understood.

The unresolved question asks for a broader classification of infinity-operads $\mathcal O$ under $\mathbb E_1$ whose algebras in spectra necessarily satisfy invariant cell numbers, thereby extending the established result beyond the $\mathbb E_4$ case.