Classification conjecture for abelian Cayley graphs with one symmetry class of Hamiltonian cycles

Let $\Gamma$ be a finite abelian group with generating set $S$, and let $G=\operatorname{Cay}(\Gamma,S)$. Let $\mathcal{H}$ denote the class of graphs having a unique Hamiltonian cycle up to symmetry. Abelian Cayley-graph classification conjecture. Then

$$G\in\mathcal{H}\quad\Longleftrightarrow\quad G\in\{C_4\mathbin{\square}K_2,\ C_n,\ K_n,\ K_{n,n},\ C_k\mathbin{\square}K_2: n,k\in\mathbb{N},\ k\text{ odd}\}.$$

This conjecture proposes a complete classification of the finite abelian Cayley graphs with one symmetry class of Hamiltonian cycles, extending the paper's results on odd-order and even-order abelian Cayley graphs. Its status is not resolved in the supplied text.