Hamiltonicity conjecture for connected bicirculants

A bicirculant is a regular graph admitting a semiregular automorphism with two vertex-orbits of equal size. Let $G(m,2)$ denote the generalized Petersen graph with parameter $m$, and let $K_2$ be the complete graph on two vertices.

Hamiltonicity conjecture. Every connected bicirculant, except for the complete graph $K_2$ and the generalized Petersen graphs $G(m,2)$ with

$$m\equiv 5\pmod 6,$$

is hamiltonian.

This conjecture seeks a complete classification of non-hamiltonian bicirculants. It extends Alspach's classification for generalized Petersen graphs and the corresponding result for $I$-graphs; the conjecture remains open in the source.