Rankin-type conjecture for Hamilton circles in two-connected Cayley graphs

Let $G=\langle S\rangle$ be a group with $S=\{a^{\pm},b^{\pm}\}$, satisfying $(ab)^2=1$, and suppose that the vertex-connectivity of $\Gamma(G,S)$ satisfies

$$\kappa(\Gamma(G,S))\geq 2.$$

A Hamilton circle is a homeomorphic image of $S^1$ in the Freudenthal compactification that contains every vertex. Rankin-type conjecture. Then $\Gamma(G,S)$ contains a Hamilton circle. This is proposed as an infinite extension of Rankin's finite Hamilton-cycle theorem; the source does not establish the conjecture.