Bonnet's sparse twin-width 3 conjecture for bounded tree-width

Let $\mathcal{C}$ be a class of graphs of twin-width at most $3$, and suppose there exists an integer $t$ such that no graph $G\in\mathcal{C}$ contains $K_{t,t}$ as a subgraph. Sparse twin-width 3 conjecture. Then $\mathcal{C}$ has bounded tree-width. This conjecture relates sparse graph classes of bounded twin-width to the structural theory of bounded tree-width; the paper proves the analogous statement for twin-width at most $2$, while the case of twin-width $3$ is presented as the motivating open problem.