  Let $\Gamma=(X,\mathcal{R})$ denote a finite, simple, connected, and
undirected non-bipartite graph with vertex set $X$ and edge set $\mathcal{R}$.
Fix a vertex $x \in X$, and define $\mathcal{R}_f = \mathcal{R} \setminus \{yz
\mid \partial(x,y) = \partial(x,z)\}$, where $\partial$ denotes the path-length
distance in $\Gamma$. Observe that the graph $\Gamma_f=(X,\mathcal{R}_f)$ is
bipartite. We say that $\Gamma$ supports a uniform structure with respect to
$x$ whenever $\Gamma_f$ has a uniform structure with respect to $x$ in the
sense of Miklavi\v{c} and Terwilliger \cite{MikTer}.
  Assume that $\Gamma$ is a distance-regular graph with classical parameters
$(D,q,\alpha,\beta)$ and diameter $D\geq 4$. Recall that $q$ is an integer such
that $q \not \in \{-1,0\}$. The purpose of this paper is to study when $\Gamma$
supports a uniform structure with respect to $x$. We studied the case $q \le 1$
in \cite{FMMM}, and so in this paper we assume $q \geq 2$. Let $T=T(x)$ denote
the Terwilliger algebra of $\Gamma$ with respect to $x$. Under an additional
assumption that every irreducible $T$-module with endpoint $1$ is thin, we show
that if $\Gamma$ supports a uniform structure with respect to $x$, then either
$\alpha = 0$ or $\alpha=q$, $\beta=q^2(q^D-1)/(q-1)$, and $D \equiv 0
\pmod{6}$.
