Aubian–Charbit–Lopes substitution conjecture for polynomially dichromatic-bounded tournaments

Let $\mathcal{C}$ be a class of tournaments, where a tournament is an orientation of a complete graph. Let $\mathcal{C}^{subst}$ denote the closure of $\mathcal{C}$ under substitution. Aubian–Charbit–Lopes substitution conjecture. If $\mathcal{C}$ is polynomially $\vec\chi$-bounded, then so is $\mathcal{C}^{subst}$. This conjecture extends the analogous graph statement that substitution preserves polynomial $\chi$-boundedness. The paper proves it for classes of tournaments with bounded dichromatic number; the general case remains open.