Injectivity of Euler-class multiplication

Determine for which representations V of the symmetric group d4a6_m multiplication by the Euler class e(\bar\sigma) of the complex standard representation is injective, and in particular prove injectivity for V=\mathbb{F}_p^{1-m}\partial_mI\otimes\mathrm{sgn}.

Background

The proposed recovery of the Arone--Mahowald cohomology calculation from the calculus version of Grays sequence requires the map Sn(k)\to\Sigma{-2}S{n+2}(k) to be injective in mod-p cohomology. This map corresponds to multiplication by e(\bar\sigma) on group cohomology with coefficients in the Lie representation V. The paper leaves the general representation-theoretic injectivity criterion, and especially the case arising from the derivatives of the identity functor, unresolved.

References

For which representations $V$ is multiplication by $e(\bar\sigma)$ injective? In particular, is there a quick proof of this injectivity for the representation $V=\mathbb{F}_p{1-m}\partial_mI\otimes \mathrm{sgn}$?

— An abelian model for the Goodwillie tower of the circle  (2609.31269 - Nervo, 25 Sep 2026) in Section 1, subsection “Further direction I: the Arone--Mahowald computations”