The Co1-equivariant non-nullhomotopy conjecture for Mathieu Moonshine

Let $V^{f\natural}$ be the holomorphic SCFT with automorphism group $\mathrm{Co}_1$, and let $\overline{\operatorname{Fer}(3)}$ be the antiholomorphic three-fermion theory. Set

$$\mathcal S=\overline{V^{f\natural}}\otimes\overline{\operatorname{Fer}(3)}.$$

The $\mathrm{Co}_1$-action on $V^{f\natural}$ induces a $\mathrm{Co}_1$-equivariant class represented by $\mathcal S$. Co1-equivariant non-nullhomotopy conjecture. The class is not $\mathrm{Co}_1$-equivariantly nullhomotopic. The paper explains that the known invariant does not obstruct the analogous $\mathrm M_{24}$-equivariant nullhomotopy, but is expected to obstruct a $\mathrm{Co}_1$-equivariant one. This remains conjectural.