Cheeger constant conjecture for distance-regular graphs

Let $G$ be a distance-regular graph, let $h_G$ denote its Cheeger constant, and let $\lambda_1$ be the smallest positive eigenvalue of the Laplacian matrix of $G$.

Cheeger constant conjecture.

$$\frac{\lambda_1}{2} \leq h_G \leq \lambda_1.$$

The lower bound is the general Cheeger inequality, while the conjectured upper bound improves the general spectral upper bound for distance-regular graphs. The conjecture is supported by results for known infinite families, diameter-two graphs, several classes of diameter-three graphs, and most distance-regular graphs with small valency.