  Juri\'{s}i\v{c} et al. conjectured that if a distance-regular graph $\Gamma$
with diameter $D$ at least three has a light tail, then one of the following
holds:
  1.$a_1 =0$;
  2.$\Gamma$ is an antipodal cover of diameter three;
  3.$\Gamma$ is tight;
  4.$\Gamma$ is the halved $2D+1$-cube;
  5.$\Gamma$ is a Hermitian dual polar graph $^2A_{2D-1}(r)$ where $r$ is a
prime power.
  In this note, we will consider the case when the light tail corresponds to
the eigenvalue $-\frac{k}{a_1 +1}$. Our main result is:
  Theorem Let $\Gamma$ be a non-bipartite distance-regular graph with valency
$k \geq 3$ , diameter $D \geq 3$ and distinct eigenvalues $\theta_0 > \theta_1
> \cdots > \theta_D$. Suppose that $\Gamma$ is $2$-bounded with smallest
eigenvalue $\theta_D = -\frac{k}{a_1 +1}$. If the minimal idempotent $E_D$,
corresponding to eigenvalue $\theta_D$, is a light tail, then $\Gamma$ is the
dual polar graph $^2A_{2D-1}(r)$, where $r$ is a prime power.
  As a consequence of this result we will also show:
  Theorem Let $\Gamma$ be a distance-regular graph with valency $k \geq 3$,
diameter $D \geq 2$, $a_1 =1$ and $\theta_0 > \theta_1 > \cdots > \theta_D$. If
$c_2 \geq5$ and $\theta_D = -k/2$, then $c_2 =5$ and $\Gamma$ is the dual polar
graph $^2A_{2D-1}(2)$.
