Subquadratic growth conjecture for acyclic subgraphs of tournaments

Let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of an oriented graph $G$. Let $g(n)$ be the smallest integer such that every tournament $G$ with more than $g(n)$ vertices has $f(G)>n$.

Subquadratic growth conjecture.

$$g(n)=o(n^2).$$

This conjecture asks for an asymptotic improvement over the general product-coloring bound $g(n)\leq n^2$. The paper proves the modest improvement $g(n)\leq n^2-(2-1/\sqrt{2})n+2$, but the conjectured subquadratic bound remains open.