Strict treewidth generalized Turán lower-bound problem

Determine whether every graph F with chromatic number r≥3 and treewidth strictly greater than r satisfies ex(n, K_r, F)=Ω(n^{r−1}).

Background

The paper disproves the previously posed assertion that every graph F with χ(F)=r≥3 and tw(F)≥r must satisfy ex(n,K_r,F)=Ω(n^{r−1}). Its counterexamples F_r have χ(F_r)=tw(F_r)=r, so they do not address the stronger regime tw(F)>χ(F). The authors therefore formulate this strengthened question. They further observe that the proposed lower bound already holds when the minimum color-class size σ(F) is at least two; consequently, any counterexample would need σ(F)=1 and would reduce to studying graphs F with a vertex v such that χ(F−v)=r−1 and tw(F−v)≥r.