Minimum-semidegree conjecture for balanced antidirected trees

Let $D$ be an oriented graph on $n$ vertices, let $\delta^0(D)$ denote its minimum semidegree, and let a balanced antidirected tree have equally many sources and sinks. The balanced-tree conjecture. Every oriented $n$-vertex graph $D$ with $\delta^0(D)>k/2$ contains every balanced antidirected $k$-edge tree whose total maximum degree is at most $o(n)$. The conjecture concerns the threshold below the trivial greedy bound and remains open.