The BIG implies BIG conjecture for tournaments

A class of tournaments $\mathcal T$ has the $BIG\Rightarrow BIG$ property if there exists a function $f$ such that, for every $T\in\mathcal T$, if $\operatorname{\overrightarrow{\chi}}(T)\geq f(t)$, then $T$ contains disjoint subtournaments $A$ and $B$ with $\operatorname{\overrightarrow{\chi}}(A),\operatorname{\overrightarrow{\chi}}(B)\geq t$ and $A\Rightarrow B$. The BIG implies BIG conjecture. The class of all tournaments has the $BIG\Rightarrow BIG$ property. This conjecture is attributed to Nguyen, Scott and Seymour and is presented as implying the Erdős–El-Zahar conjecture.