Asymptotic oriented discrepancy conjecture for directed rooted trees

Let $T$ be a tree with $\ell$ leaves, and let $\mathcal{DT}$ be the set of all directed rooted trees, namely trees with a distinguished root and all edges oriented away from it. Write $\vec{\mathcal{D}}(T,\mathcal{DT})$ for the oriented discrepancy of this family in $T$. Asymptotic oriented discrepancy conjecture. For every such tree,

$$\vec{\mathcal{D}}(T,\mathcal{DT})=\left(1+o(1)\right)\frac{\ell}{2}.$$

The proven lower bound is $\lceil\ell/2\rceil+1$, while the current general upper bound is $\ell$; the conjecture asks for an asymptotically matching upper bound.