Triviality conjecture for the Ekedahl invariants of the p-discrete Heisenberg group

Let $p$ be a prime and let $H_p$ denote the $p$-discrete Heisenberg group. Its Ekedahl invariants are the invariants $\operatorname{e}_i(H_p)$ associated with the classifying space of $H_p$. Triviality conjecture. The Ekedahl invariants of $H_p$ are trivial; that is, $\operatorname{e}_i(H_p)=0$ for every index $i$. This conjecture extends the known result for the fifth discrete Heisenberg group, while the general case remains open because of the technical difficulties in extending the relevant result beyond $p=5$.