Georgakopoulos–Mohar conjecture on Hamilton circles in planar Cayley graphs

Let $\Gamma$ be a finitely generated 3-connected planar Cayley graph. A Hamilton circle is a homeomorphic image of $S^1$ in the Freudenthal compactification of the graph that contains every vertex. Georgakopoulos–Mohar conjecture. Every finitely generated 3-connected planar Cayley graph admits a Hamilton circle. This is the broader conjecture for which the preceding three-involution problem is proposed as a first step; the source gives no proof or resolution.