Burr's conjecture for oriented graphs

Establish that every oriented graph with chromatic number at least $2k-2$ contains every oriented tree on $k$ vertices; equivalently, prove that the threshold function $f(k)$, defined as the smallest integer such that every oriented graph of chromatic number at least $f(k)$ contains every oriented tree on $k$ vertices, satisfies $f(k)=2k-2$.

Background

For an oriented graph, the chromatic number is taken to be the chromatic number of its underlying undirected graph. The function $f(k)$ measures the minimum chromatic threshold forcing every oriented tree on $k$ vertices to occur as a subgraph. Burr proved the quadratic upper bound $f(k)\le (k-1)^2$ and the paper notes the lower bound $f(k)\ge 2k-2$, obtained from a regular tournament on $2k-3$ vertices, whose maximum out-degree is $k-2$ and therefore does not contain the out-star on $k$ vertices.

The paper improves the best previously known general upper bound to $f(k)=O(k\log k)$, leaving a factor of order $O(\log k)$ between the known upper and lower bounds. Thus the conjectured exact value remains unresolved in the general setting.