Determine the sharp number of cycle sets

Determine how far the upper bound that the number of cycle sets of n-vertex graphs is at most 2^{n - Ω(√n / log^{3/2} n)} is from the true asymptotic maximum, in view of the known lower bound of 2^{n/2}.

Background

The paper improves Verstraëte’s upper bound on the number of distinct cycle sets of n-vertex graphs to 2^{n - Ω(√n / log^{3/2} n)}. It contrasts this with a construction due to Faudree yielding at least 2^{n/2} distinct cycle sets. The authors explicitly state that the distance between these bounds is unknown, leaving the sharp asymptotic enumeration problem unresolved.