The authors prove the bicirculant Hamiltonicity conjecture for |S| <= 3 and for |S| >=4 with m/gcd(m,S) even, plus a reduction showing that Hamiltonicity of valence >=4 cyclic Haar graphs implies it for all bicirculants of valence >=4.
Maruˇ siˇ c, On vertex symmetric digraphs,Discrete Math.36(1981), 69–81
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On the hamiltonicity problem of bicirculants: a reduction to cyclic Haar graphs
The authors prove the bicirculant Hamiltonicity conjecture for |S| <= 3 and for |S| >=4 with m/gcd(m,S) even, plus a reduction showing that Hamiltonicity of valence >=4 cyclic Haar graphs implies it for all bicirculants of valence >=4.