Hamiltonicity conjecture for two-connected Cayley graphs generated by three involutions

Let $G$ be a group generated by three involutions $a,b,c$ such that $ab=ba$, and let $\Gamma(G,\{a,b,c\})$ be its Cayley graph. Three-involution conjecture. If $\Gamma(G,\{a,b,c\})$ is two-connected, then it is Hamiltonian. The conjecture is proposed as a first step toward a broader conjecture on Hamilton circles in 3-connected planar Cayley graphs; its resolution is not given in the source.