On a characterization of the Grassmann graphs

In 1995, Metsch showed that the Grassmann graph $J_q(n,D)$ of diameter $D\geq 3$ is characterized by its intersection numbers with the following possible exceptions: (-) $n=2D$ or $n=2D+1$, $q\ge 2$; (-) $n=2D+2$ and $q\in \{2,3\}$; (-) $n=2D+3$ and $q=2$. In 2005, Van Dam and Koolen constructed the twisted Grassmann graphs with the same intersection numbers as the Grassmann graphs $J_q(2D+1,D)$, for any prime power $q$ and diameter $D\ge 2$, but they are not isomorphic. We show that the Grassmann graph $J_q(2D,D)$ is characterized by its intersection numbers provided that the diameter $D$ is large enough.