The good ordering conjecture for tournament twin-width and clique number

For a tournament $T$ and an ordering $\prec$ of $V(T)$, let $T^{\prec}$ be its backedge graph. A BST-ordering is the ordering associated with a binary search tree satisfying the tournament-neighbourhood conditions described in the paper. The good ordering conjecture. There exists a function $f$ such that, for every tournament $T$, there exists an ordering $\prec^*$ of $V(T)$ such that

$$\omega(T^{\prec^*})\leq f(\operatorname{\overrightarrow{\omega}}(T))\quad\text{and}\quad tww(T,\prec^*)\leq f(tww(T)).$$

If true, this would imply the bounded twin-width conjecture; the paper notes that BST-orderings are natural candidates because they already provide the required twin-width bound, leaving the backedge-graph clique bound open.