Bounding the parameter $\beta$ of a distance-regular graph with classical parameters

Let $\Gamma$ be a distance-regular graph with classical parameters $(D, b, \alpha, \beta)$ satisfying $b\geq 2$ and $D\geq 3$. Let $r=1+b+b^2+\cdots+b^{D-1}$. In 1999, K. Metsch showed that there exists a positive constant $C(\alpha,b)$ only depending on $\alpha$ and $b$, such that if $\beta \geq C(\alpha, b)r^2$, then either $\Gamma$ is a Grassmann graph or a bilinear forms graph. In this work, we show that for $b\geq 2$ and $D\geq 3$, then there exists a constant $C_1(\alpha, b)$ only depending on $\alpha$ and $b$, such that if $\beta \geq C_1(\alpha, b)r$, then either $\Gamma$ is a Grassmann graph, or a bilinear forms graph.