Equality of the Brun–Willwacher and Kupers–Miller–Patzt sign-representation classes

Determine whether the classes \(\nu_{2k}^{\mathrm{BW}}\) and \(\nu_{2k}^{\mathrm{KMP}}\) are equal up to a multiplicative constant for all relevant \(k\).

Background

The paper discusses two families of homology classes with coefficients in the sign representation, $\nu_{2k}^{\mathrm{BW}}$ and $\nu_{2k}^{\mathrm{KMP}}$, introduced in earlier work. The $\nu_{2k}^{\mathrm{KMP}}$ classes are known to be nontrivial for $k\geq 2$, while the corresponding nontriviality statement for the Brun–Willwacher classes is said to follow by the same argument, with details deferred to future work. The relationship between the two families is explicitly presented as conjectural.