Lee, Loh and Sudakov's directed-cut bisection conjecture

Let $D$) be a digraph with $m$ arcs, and for a bipartition $V(D)=V_1\cup V_2$, let $e(V_i,V_j)$ denote the number of arcs directed from $V_i$ to $V_j$. Let $d\geq 3$ be a positive integer.

Lee, Loh and Sudakov's conjecture. Every digraph $D$ with minimum outdegree at least $d+1$ admits a bipartition $V(D)=V_1\cup V_2$ such that

$$\min\left\{e(V_1,V_2),e(V_2,V_1)\right\}\geq \left(\frac{d}{2(2d+1)}+o(1)\right)m.$$

This conjecture extends the known asymptotic cases $c_2=1/6+o(1)$ and $c_3=1/5+o(1)$ for the largest guaranteed proportion of arcs in both directions of a directed cut. The general case remains open.