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

Let $\Gamma$ be a geometric distance-regular graph with diameter $D \geq 3$ and distinct eigenvalues

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

Let $b = \frac{b_1}{\theta_1 + 1}$. For vertices $x,y$ at distance $i$, let $\tau_i$ be the number of Delsarte cliques containing $y$ and a vertex at distance $i-1$ from $x$. The $\tau_i$-bound conjecture. There exists a function $f(i,b)$ such that $\tau_i \leq f(i,b)$. This conjecture would generalize the problem of bounding the smallest eigenvalue in terms of $D$ and $b$, since $\theta_D=-\tau_D$ for geometric distance-regular graphs; its status is open.