Lee–Loh–Sudakov conjecture on judicious bipartitions of digraphs

For a digraph $D$, let $V(D)$ be its vertex set, let $m$ be its number of arcs, and let $e(A,B)$ denote the number of arcs directed from $A$ to $B$. A bipartition is a decomposition $V(D)=V_1\cup V_2$. Let $d$ be an integer satisfying $d\geq 4$.

Lee–Loh–Sudakov conjecture. Every digraph $D$ with $m$ arcs and minimum outdegree at least $d$ admits a bipartition $V(D)=V_1\cup V_2$ with

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

This conjecture asks for the optimal asymptotic lower bound on the smaller of the two directed cut sizes in a digraph with prescribed minimum outdegree. Lee, Loh, and Sudakov proved the corresponding asymptotic values for minimum outdegree $2$ and $3$; the case $d\geq 4$ is presented as open here.