Authors' classification conjecture for non-bipartite distance-regular graphs

Let $\Gamma$ be a non-bipartite distance-regular graph with valency $k$, diameter $D$, and smallest eigenvalue $\theta_{\min}\leq -k/2$. The graph is assumed to have sufficiently large diameter.

Authors' classification conjecture. When $D$ is large enough, $\Gamma$ is one of the following graphs: the odd polygons; folded $(2D+1)$-cubes; the odd graphs $O_k$; the Hamming graphs $H(D,3)$; the dual polar graphs $^2A_{2D-1}(2)$; or the dual polar graphs $B_D(2)$.

This conjecture proposes a classification of non-bipartite distance-regular graphs under a smallest-eigenvalue bound. The source notes that it is correct when both $c_2\geq 3$ and $a_1=1$ hold, while the general large-diameter case remains open.