Asymptotic equivalence of graph and digraph circumferences

Let $c(n)$ be the minimum circumference of a connected vertex-transitive graph on $n$ vertices, and let $d(n)$ be the minimum circumference of a connected vertex-transitive digraph on $n$ vertices. Asymptotic circumference conjecture.

$$d(n)=\Theta(c(n)).$$

The paper proves a lower bound of order $n^{1/3}$ for the directed circumference and notes that matching the best known undirected bounds, or showing asymptotic agreement between the directed and undirected cases, would be interesting; the supplied source does not indicate whether this conjecture has been resolved.