Cubic arc-transitive k-multicirculants

For an integer k⩾1, a graph is called a k-multicirculant if its automorphism group contains a cyclic semiregular subgroup with k orbits on the vertices. If k is even, there exist infinitely many cubic arc-transitive k-multicirculants. We conjecture that, if k is odd, then a cubic arc-transitive k-multicirculant has order at most 6k2. Our main result is a proof of this conjecture when k is squarefree and coprime to 6.