Dallard et al.'s equivalence conjecture for tree-independence number

Let $\mathcal{G}$ be a hereditary graph class. A graph class is $(\mathsf{tw},\omega)$-bounded if its treewidth is bounded by a function of its clique number, and it has bounded tree-independence number if the tree-independence numbers of its graphs are uniformly bounded. Dallard et al.'s conjecture. The class $\mathcal{G}$ is $(\mathsf{tw},\omega)$-bounded if and only if $\mathcal{G}$ has bounded tree-independence number. Bounded tree-independence number is known to imply $(\mathsf{tw},\omega)$-boundedness, and the converse remains open in general; it is established for several important graph classes.