Rappaport–Strasser conjecture for Hamiltonian Cayley graphs

Given a finite group $G$ and a symmetric subset $S\subset G$, the Cayley graph $\operatorname{Cay}_G(S)$ has vertex set $G$ and edges $\{g,gs\}$ for $g\in G$ and $s\in S$. A graph is Hamiltonian if it contains a cycle through every vertex. Rappaport–Strasser conjecture. Every connected Cayley graph on a finite group with at least $3$ elements is Hamiltonian. This is the Cayley-graph version of Lovász's conjecture, posed in 1959, and remains wide open despite results for abelian groups and other special families.