Dallard–Milanič–Štorgel tree-independence conjecture for hereditary classes

Let $\mathcal{G}$ be a hereditary class of graphs. A class has bounded tree-independence number if there is a constant $c\in\mathbb{N}$ such that every graph in the class admits a tree decomposition whose bags induce subgraphs with no stable set on $c+1$ vertices; it is $(\operatorname{{\sf tw}},\omega)$-bounded if there is a function $f:\mathbb{N}\to\mathbb{N}$ such that $\operatorname{{\sf tw}}(G)\leq f(\omega(G))$ for every graph $G$ in the class. The tree-independence number is denoted by $\operatorname{{\text{\sf{tree}}-\alpha}}(G)$. Dallard–Milanič–Štorgel's conjecture. A hereditary class $\mathcal{G}$ has bounded $\operatorname{{\text{\sf{tree}}-\alpha}}$ if and only if $\mathcal{G}$ is $(\operatorname{{\sf tw}},\omega)$-bounded. Chudnovsky and Trotignon refuted this conjecture: there are hereditary $(\operatorname{{\sf tw}},\omega)$-bounded classes whose bounding functions can grow arbitrarily fast, so the asserted equivalence fails.