A bound on the clique parameters of geometric distance-regular graphs

Let $\Gamma$ be a geometric distance-regular graph with diameter $D$ and distinct eigenvalues

$$k = \theta_0 > \theta_1 > \cdots > \theta_D.$$

Let $b = \frac{b_1}{\theta_1 + 1}$. For a Delsarte clique, let $\phi_i$ denote the number of clique vertices at distance $i$ from a vertex at distance $i$ from the clique. The $\phi_i$-bound conjecture. There exists a function $f(i,b)$ such that, if $D \geq 2i + 1$, then $\phi_i \leq f(i,b)$. This conjecture seeks uniform bounds on the clique parameters of geometric distance-regular graphs; the preceding proposition establishes such a bound for $\phi_1$ under an additional lower bound on $\frac{k}{-\theta_D}$, while the general assertion remains open.