In this paper we show that for integers [Formula presented], [Formula presented], any co-edge-regular graph which is cospectral with the [Formula presented]-clique extension of the [Formula presented]-grid is the [Formula presented]-clique extension of the [Formula presented]-grid, if [Formula presented] is large enough. Gavrilyuk and Koolen used a weaker version of this result to show that the Grassmann graph [Formula presented] is characterized by its intersection array as a distance-regular graph, if [Formula presented] is large enough.