Dallard–Krnc–Kwon–Milanič–Munaro–Štorgel–Wiederrecht conjecture for finitely forbidden induced subgraphs

Let $\mathcal{F}$ be a finite class of graphs. A graph is $\mathcal{F}$-free if it has no induced subgraph isomorphic to any member of $\mathcal{F}$. The class of all $\mathcal{F}$-free graphs 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 such graph; bounded $\operatorname{{\text{\sf{tree}}-\alpha}}$ means that their tree decompositions have bags with bounded independence number. Dallard–Krnc–Kwon–Milanič–Munaro–Štorgel–Wiederrecht's conjecture. The class of all $\mathcal{F}$-free graphs has bounded $\operatorname{{\text{\sf{tree}}-\alpha}}$ if and only if it is $(\operatorname{{\sf tw}},\omega)$-bounded. This is a restriction of the broader hereditary-class conjecture and is presented as a conjecture before being proved in the paper.