Gutowski–Rams explicit tournament family conjecture

An explicit family of tournaments is a sequence $(T_n)_{n\geq 1}$ of tournaments, with $|V(T_n)|$ denoting the number of vertices and $\vec\omega(T_n)$ denoting the clique number. Gutowski–Rams conjecture. There exists an explicit family of tournaments $(T_n)_{n\geq 1}$ such that

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

and

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

The conjecture concerns constructing tournaments whose order is linear in the parameter while their clique number grows polynomially. The paper states that its method gives a positive answer to this conjecture, so the claim is treated as solved.