Havet's chromatic conjecture for trees with few leaves

Let $k\in\mathbb N$, let $\ell\in\mathbb N$, let $D$ be a digraph, and let $T$ be an oriented tree with $k$ edges and $\ell$ leaves. Havet's conjecture. Every $(k+\ell)$-chromatic digraph contains every oriented $k$-edge tree having $\ell$ leaves. This generalizes Burr's chromatic conjecture and the tournament conjecture for trees with few leaves; it is known only in restricted cases.