Rapaport–Strasser conjecture for Hamilton cycles in connected Cayley graphs

Let $\Gamma$ be a finite group and let $S\subseteq\Gamma\setminus\{1\}$ satisfy $S=S^{-1}$. The Cayley graph $\operatorname{Cay}(\Gamma,S)$ has vertex set $\Gamma$ and edges $\{g,gs\}$ for $g\in\Gamma$ and $s\in S$; assume it is connected, equivalently, that $S$ generates $\Gamma$. A Hamilton cycle is a cycle containing every vertex of the graph.

Rapaport–Strasser conjecture. Every connected Cayley graph on a finite group with at least three elements has a Hamilton cycle.

The conjecture is known for finite abelian groups and for several dense classes of Cayley graphs, but the full conjecture remains open.