The complete-pairs formulation of the Gyárfás–Sumner conjecture

Let $G$ be a finite simple graph. For disjoint vertex sets $A,B\subseteq V(G)$, call $(A,B)$ a complete pair if every possible edge between $A$ and $B$ is present. For a forest $T$, call $G$ $T$-free if it has no induced subgraph isomorphic to $T$. Complete-pairs formulation. For every $\ell,w\ge1$ and every forest $T$, there exists $k\ge2$ such that every $T$-free graph $G$ with $\chi(G)\ge k$ and $\omega(G)\le w$ contains a complete pair $(A,B)$ with $\chi(A),\chi(B)\ge\ell$. The source states that this is equivalent to the Gyárfás–Sumner conjecture, so it is another formulation of the same open problem.