Hamiltonicity conjecture for one-ended generalized quasi-dihedral groups

Let $G=\langle S\rangle$ be a one-ended generalized quasi-dihedral group, and let $\operatorname{Cay}(G,S)$ denote its Cayley graph. A Hamiltonian double ray is a spanning double ray, while a Hamiltonian circle is a circle in the Freudenthal compactification of the graph that contains all its vertices. One-ended generalized quasi-dihedral Hamiltonicity conjecture. The Cayley graph $\operatorname{Cay}(G,S)$ contains a Hamiltonian double ray or a Hamiltonian circle. The paper establishes Hamiltonian double rays and, in the two-ended setting, Hamiltonian circles for the relevant groups; it proposes that the analogous Hamiltonicity statements also hold for one-ended generalized quasi-dihedral groups, where the index-two abelian subgroup is isomorphic to $\mathbb Z^k$ for some $k\geq 2$.