Asymptotic order conjecture for the clique ratio of regular graphs with bounded smallest eigenvalue

For an integer $\lambda\geq 2$, let $\gamma(\lambda)$ be the limiting infimum of $\omega(G)/k$ as $k$ tends to infinity over connected $k$-regular graphs $G$ with smallest eigenvalue at least $-\lambda$, where $\omega(G)$ is the clique number of $G$:

$$\gamma(\lambda):=\lim_{k\to\infty}\inf\left\{\frac{\omega(G)}{k}\mid G\text{ is a connected }k\text{-regular graph and }\lambda_{\min}(G)\geq-\lambda\right\}.$$

Asymptotic clique-ratio conjecture. There exist positive constants $\varepsilon_1$ and $\varepsilon_2$ such that

$$\frac{\varepsilon_1}{\lambda^2}\leq\gamma(\lambda)\leq\frac{\varepsilon_2}{\lambda^2}$$

holds for every integer $\lambda$. The preceding bounds show that the authors seek to improve their lower bound, while Hamming graphs provide an upper bound of order $1/\lambda$; the conjecture predicts the sharper order $1/\lambda^2$.