Approximation conjecture for the ordered clique number of tournaments

Let $T$ be a tournament, and let $T^{\prec}$ be the graph associated with an ordering $\prec$ of $V(T)$ by retaining the backward arcs. Define the ordered clique number by

$$\overrightarrow{\omega}(T)=\min\{\omega(T^{\prec}):\prec\text{ is a total order of }V(T)\}.$$

Approximation conjecture. There is a function $f$ such that for every integer $k$, there is a polynomial-time algorithm which, given a tournament $T$, correctly concludes that $\overrightarrow{\omega}(T)\geq k$, or finds an ordering $\prec$ of $V(T)$ such that $\omega(T^{\prec})\leq f(k)$. This conjecture proposes a polynomial-time gap approximation for the ordered clique number; the source states preceding approximation results but gives no resolution of the conjecture.