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.
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CSI of Charles University
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.