Bollobás–Scott weighted long-cycle conjecture for Eulerian digraphs

A digraph is a directed graph; it is Eulerian when it is strongly connected and every vertex has equal in-degree and out-degree. For a digraph $G$, let $A(G)$ denote its arc set, let $w:A(G)\to\mathbb{R}_{\geq 0}$ be a weight function, and write $w(G)$ for the total weight of its arcs. Bollobás–Scott's weighted long-cycle conjecture. Every $n$-vertex Eulerian digraph $G$ with a weight function $w:A(G)\to\mathbb{R}_{\geq 0}$ has a directed cycle of length at least

$$c\cdot\frac{w(G)}{n}$$

for some universal constant $c>0$. This conjecture proposes a linear lower bound in the average arc weight for the length of a directed cycle and generalizes unweighted long-cycle questions for Eulerian digraphs.