The book you link to has examples of directed Cayley graphs with no (directed) Hamiltonian cycle. The existence of such graphs is interesting and relevant, but does not give a counterexample to the problem stated here.

## graphs vs. digraphs

The book you link to has examples of directed Cayley graphs with no (directed) Hamiltonian cycle. The existence of such graphs is interesting and relevant, but does not give a counterexample to the problem stated here.