Explicit polynomial directed-clique construction conjecture for tournaments

Let $T_1,T_2,\ldots$ be tournaments, and write $V(T_n)$ for the vertex set of $T_n$. Explicit tournament construction conjecture. There is an explicit construction of tournaments $T_1,T_2,\ldots$ such that

$$|V(T_n)|=\Theta(n)$$

and

$$\operatorname{\overrightarrow{\omega}}(T_n)=\Omega(n^{1/100}).$$

Such a construction would substantially improve the currently discussed logarithmic lower bounds for directed clique number in tournaments; its existence is left open by the source.