Additional endopure transfers for odd extraspecial p-groups

Determine whether, for each odd prime p and each n, the transfer $\Omega(p_+^{1+2(n-1)}\circ C_{p^2})\cong p_+^{1+2(n-1)}\lneq (p_+^{1+2(n-1)}\circ C_{p^2})$ is the only new endopure transfer in the family of extraspecial p-groups of order $p^{1+2n}$ and their subgroups.

Background

The paper proves that the family of all extraspecial p-groups for an odd prime p has infinitely many global transfer systems, using the endopure transfers arising from the groups $p_-^{1+2n}$. For the finite subfamilies consisting of groups of a fixed order parameter n, the complete collection of endopure transfers and the resulting lattice structure are not determined.

Computations identify one further endopure transfer, namely the transfer from the elementary-exponent extraspecial factor to its central product with $C_{p^2}$. The authors conjecture that this is the only additional transfer; if so, the associated lattice would be a finite chain of length $2n+1$.