Tournament Gyárfás–Sumner conjecture for forest backedge graphs

Let $T$ be a tournament with an ordering of its vertices, and let the corresponding backedge graph be the graph whose edges are the arcs directed backwards in that ordering. A tournament class is dichromatically bounded by clique number if there is a function $f$ such that every tournament in the class satisfies

$$\operatorname{\overrightarrow{\chi}}(T)\leq f(\operatorname{\overrightarrow{\omega}}(T)).$$

Tournament Gyárfás–Sumner conjecture. If $T$ has a backedge graph that is a forest, then the class of tournaments not containing $T$ as a subgraph is dichromatically bounded by clique number. The source states this as an analogue of the Gyárfás–Sumner conjecture and does not give a resolution here.