A diameter-bound conjecture for distance-regular graphs with classical parameters

Let $\Gamma$ be a distance-regular graph with classical parameters $(D,b,\alpha,\beta)$.

Diameter-bound conjecture. There is a constant $\ell$ such that if $D\geq\ell$, then one of the following statements holds:

1. If $b$ is a perfect square, then $\alpha\leq b+\sqrt{b}$.
2. If $b$ is not a perfect square, then $\alpha\leq b$.

This conjecture proposes a uniform bound on the parameter $\alpha$ for sufficiently large diameter, with the exceptional value $b+\sqrt{b}$ allowed only when $b$ is a perfect square. The paper establishes related bounds for restricted ranges of $b$ and large $D$, but the stated general assertion remains open.