Characterization of endopure transfers in finite extraspecial p-groups

Determine whether the explicitly identified endopure transfers exhaust all endopure transfers in the subgroups of the odd extraspecial p-groups $p_-^{1+2n}$, equivalently, determine whether subgroups of $p_-^{1+2n}$ other than the identified examples possess endopure transfers.

Background

The paper reduces the unresolved classification problem for subgroups of $p_-^{1+2n}$ to subgroups of exponent $p^2$, because groups of exponent p have no proper endopure subgroups except the cyclic group $C_p$.

The authors specifically ask whether there are endopure transfers beyond the transfers already constructed, making the classification of meet- and join-irreducible global transfer systems for the finite extraspecial-family subcategories unresolved.