Bollobás–Scott linear long-cycle conjecture for Eulerian digraphs

A digraph is Eulerian when it is strongly connected and every vertex has equal in-degree and out-degree. Its average out-degree is the average of the out-degrees of its vertices; denote this quantity by $d$. Bollobás–Scott's linear long-cycle conjecture. Every Eulerian digraph with average out-degree $d$ has a directed cycle of length at least

$$cd$$

for some universal constant $c>0$. This is a weaker unweighted version of the weighted Bollobás–Scott conjecture and was proposed as an intermediate step toward resolving it; the source does not state a resolution.