The elementary-abelian Chern-class bound

Let $C$ be an elementary abelian $p$-group of rank $c$, and let $\rho$ be a faithful $n$-dimensional complex representation of $C$. Let $e(\rho)$ be the top degree of the indecomposables of $H^*(BC)$ over the Hopf algebra generated by the Chern classes of $\rho$. Elementary-abelian Chern-class bound.

$$e(\rho) \leq 2n-c.$$

The paper states that this conjecture would immediately imply the compact-Lie-group Chern-class upper bound. It concerns the Chern classes of faithful representations of elementary abelian groups and is used as the starting point for a further conjectural identification of the associated Hopf algebra.