Hamiltonian circle conjecture for two-ended Cayley graphs with cyclic prime-power commutator subgroup

Let $G=\langle S\rangle$ be a two-ended group, and suppose that its commutator subgroup satisfies

$$G'\cong \mathbb{Z}_{p^n}.$$

The Hamiltonian circle conjecture. The Cayley graph

$$\operatorname{Cay}(G;S)$$

has a hamiltonian circle. This extends the preceding result for one-ended Cayley graphs with $G'\cong\mathbb{Z}_p$; the conjecture concerns the existence of Hamiltonian circles in the unresolved two-ended case.