Minimum-pseudo-semidegree conjecture for balanced antidirected trees

Let $D$ be an oriented graph on $n$ vertices, let $\bar\delta^0(D)$ denote its minimum pseudo-semidegree, and let a balanced antidirected tree have equally many sources and sinks. The pseudo-semidegree balanced-tree conjecture. For each $k\in\mathbb N$, every oriented graph on $n$ vertices with $\bar\delta^0(D)>k/2$ contains every balanced antidirected tree with $k$ edges and maximum total degree $o(n)$. This is proposed as an exact, non-asymptotic-strengthening of known approximate results and remains open.