Conjectured sharp clique-density bound for graphs with no short holes

Prove that for every integer $g\\geq 3$, every $2$-Leray graph over a field $\\Bbbk$ with no holes of lengths in $[4,g]$ and edge density $\\varepsilon\\in(2/g,1)$ has asymptotic clique-number density at least $1-\\sqrt{\\frac{(g-1)^2}{g^2-2g}\\sqrt{1-\\varepsilon}}$.

Background

For $2$-Leray graphs with no holes of length at most gg, the paper proves a lower bound involving the factor g/(g−2)g/(g-2). Joins of circulant graphs suggest a stronger factor (g−1)2/(g2−2g)(g-1)^2/(g^2-2g), and the authors explicitly formulate this sharper asymptotic estimate as a conjecture for edge densities above $2/g$.

References

Let $g \geq 3$ be an integer. For $2$-Leray graphs over a field $\Bbbk$ with no holes in the range $[4,g]$, if the edge density $\varepsilon \in (\frac{2}{g}, 1)$, the clique density is asymptotically (as $n \to \infty$) bounded by \ \kappa \geq 1 - \sqrt{\frac{(g-1)2}{g2-2g} \sqrt{1-\varepsilon}. \

— Clique number and triangle densities in $C_4$-free graphs  (2608.19686 - Fløystad et al., 20 Aug 2026) in Section “Further discussion,” paragraph “Forbidding longer holes,” displayed Conjecture